Robust Communication in Vehicular Ad Hoc Networks

Mark Johnson

Electrical Engineering and Computer SciencesUniversity of California at Berkeley

Technical Report No. UCB/EECS-2010-162

http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-162.html

December 16, 2010

Copyright © 2010, by the author(s).All rights reserved.

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Robust Communication in Vehicular Ad Hoc Networks

by

Mark Christopher Johnson

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Engineering - Electrical Engineering and Computer Sciences

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Kannan Ramchandran, ChairProfessor Michael GastparProfessor Raja Sengupta

Fall 2010

Robust Communication in Vehicular Ad Hoc Networks

Copyright 2010by

Mark Christopher Johnson

1

Abstract

Robust Communication in Vehicular Ad Hoc Networks

by

Mark Christopher Johnson

Doctor of Philosophy in Engineering-Electrical Engineering and Computer Sciences

University of California, Berkeley

Professor Kannan Ramchandran, Chair

The development of intelligent transportation systems is one of the most important strate-gies for making roads safer and more efficient. The key technology underpinning all suchsystems, from cooperative safety systems to electronic tolling, is wireless communications.Vehicles must be able to communicate with each other, and with fixed infrastructure, in realtime. One promising architecture involves forming an ad hoc network among the vehiclesand taking advantage of the peer-to-peer connectivity to reduce infrastructure costs.

In this dissertation, we consider the design and analysis of medium access control andnetwork layer protocols for vehicular ad hoc networks. In particular, we examine the problemof distributing content over many hops from a source to a destination. The fundamentalchallenge in such systems is the inability of the vehicles to coordinate with each other.Because of the high frequency at which vehicles enter and exit the road, and the fact thatvehicles may be moving at widely varying speeds, the topology is continuously changing.Therefore, centralized scheduling of transmissions to avoid collisions and global routing ofpackets is either technically infeasible or prohibitively expensive. We propose instead a classof uncoordinated protocols, based on randomized channel access and network coding. Thekey idea behind the coding is for the vehicles to transmit random combinations of all of thepackets that they have received, instead of simply forwarding packets. This scheme doesnot require link level feedback, and thus vehicles do not need to track the identities of theircurrent neighbors.

We first consider a minimally coordinated protocol, where the vehicles on a road arepartitioned into spatial blocks. The block structure is used to funnel waves of data downthe road, while the vehicles within each block randomly contend with each other for thechannel. We show that when the vehicles use network coding in this scheme, the end-to-end throughput is constant, as long as the number of blocks in the network does not growexponentially in the vehicle density. Network coding allows for the reliable transport of dataover many hops at the same rate at which data could be sent over a single hop.

Then we consider a completely uncoordinated protocol, which does not even requirethe assignment of vehicles to spatial blocks. We begin by providing a general algorithm forcomputing the throughput of an arbitrary network using such a protocol. The main technicalcontribution is to account for the fact that intermediate nodes may temporarily not have

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any new packets to transmit to their neighbors, due to the random channel access. We provethat the throughput of this wireless network is equal to the throughput of an equivalentwired network, despite the randomized medium access strategy. This result has applicationin a variety of other contexts, where it is often assumed for tractability that relays alwayshave data when they use the channel.

We then apply this result to the problem of finding the throughput in an uncoordinatedvehicular ad hoc network. In a congested highway, where the connectivity graph is homoge-neous, a constant throughput can be provided to receivers that are located at an arbitrarydistance from the source. In essence, every vehicle can be thought of as a digital repeater,despite the packet losses due to collisions.

i

To my parents and sister

ii

Contents

List of Figures iii

List of Tables vii

1 Introduction 1

2 Multihop Content Distribution in Vehicular Ad Hoc Networks 5

2.1 Multihop Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 MAC Layer Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Network Layer Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 A Minimally Coordinated VANET Communication Protocol 13

3.1 Spatial Partitioning of the Road . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Throughput Of Synchronous Protocol . . . . . . . . . . . . . . . . . . . . . . 163.3 Synchronous Transmission with Overhearing . . . . . . . . . . . . . . . . . . 233.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Uncoordinated Wireless Networks 29

4.1 Neighbor-Oblivious Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 A Distributed Communication Protocol for Neighbor-Oblivious Networks . . 304.3 Capacity of Neighbor-Oblivious Networks . . . . . . . . . . . . . . . . . . . . 32

5 An Uncoordinated VANET Communication Protocol 42

5.1 Modeling the Vehicular Network Connectivity . . . . . . . . . . . . . . . . . 435.2 Throughput of A Dense Vehicular Network . . . . . . . . . . . . . . . . . . . 445.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Conclusions and Future Directions 59

Bibliography 62

iii

List of Figures

1.1 A single hop vehicular network. The vehicles communicate only with the fixedbase stations along the roadside. . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 A vehicular ad hoc network. The vehicles communicate directly with eachother, without requiring a base station to organize and manage the network. 3

2.1 We consider a unicast communication problem in a multihop wireless vehicularad hoc network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 In a system based on random linear network coding, the intermediate nodestransmit linear combinations of the packets that they have received. . . . . . 10

3.1 In the synchronous transmission scheme, the highway is partitioned into blocks,where the length of each block is half of the transmission range of the vehicles. 14

3.2 In the synchronous transmission scheme, the time axis is divided into slots,which are then grouped into frames composed of M slots. . . . . . . . . . . . 14

3.3 In the synchronous scheme, waves of data propagate along the highway, mov-ing one block to the right in each frame. The dark arrows show two spatialblocks that are transmitting during the same frame. The concurrent wavesare sufficiently separated so that there is no interference between them. . . . 15

3.4 A random graph modeling the propagation of a wave of data through a ve-hicular ad hoc network using the synchronous transmission scheme. Eachnode, and all of its incoming and outgoing edges, is independently deletedwith probability 1 − 1/e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 An instantiation of the random graph modeling the propagation of a sin-gle wave through the network. The white circles represent nodes that weredeleted, and the black circles represent nodes that were not deleted. Thisgraph has only a single vertex-disjoint end-to-end path. One such path isoutlined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 An instantiation of the random graph modeling the propagation of a sin-gle wave through the network. The white circles represent nodes that weredeleted, and the black circles represent nodes that were not deleted. Thisgraph has r = 3 vertex-disjoint end-to-end paths. One set of three disjointpaths is outlined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

iv

3.7 The number of independent packets received per wave as a function of thelocation of the destination vehicle in the synchronous transmission scheme.Results are shown for four different strategies: routing with the basic syn-chronous protocol, routing with overhearing, network coding with the basicsynchronous protocol, and network coding with overhearing. . . . . . . . . . 26

3.8 The total number of received packets per wave as a function of the locationof the destination vehicle in the synchronous transmission scheme. This plotcounts all packets received without collision, regardless of whether they areindependent. Results are shown for four different strategies: routing with thebasic synchronous protocol, routing with overhearing, network coding withthe basic synchronous protocol, and network coding with overhearing. . . . . 27

3.9 The time when the infostation begins transmitting each wave, and the timewhen the destination vehicle receives the final packet of the wave, for thescenario outlined in Table 3.1. The vertical distance between the two curvesis the time delay of the synchronous scheme. . . . . . . . . . . . . . . . . . . 28

4.1 We consider a unicast communication problem in a multihop neighbor-obliviousnetwork. Our objective is to find the maximum throughput under the limita-tion that none of the nodes know the identities of their neighbors. . . . . . . 30

4.2 This simple network illustrates the starvation problem that can occur whennodes access the channel in a random order. Consider a case when the sourceand relay transmit in the order S, R, R. When the relay transmits the secondtime, it has not yet received a second packet from the source and hence canonly send a redundant packet to D. . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 A linear wireless network with four links. The independent sets, which aresets of links that can be used concurrently without interfering with each other,are {1}, {2}, {3}, {4}, and {1, 4}. . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 The set of links that are currently active can be modeled with a Markovchain defined on the independent sets. This figure shows the Markov chaincorresponding to the network in Figure 4.3 when using a carrier sensing MACprotocol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 The wireless network in Figure 4.3 can be replaced with an equivalent wirednetwork, know as a link capacity graph, whose individual links have the samecapacities as the wireless links. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.6 A diamond network with source S, destination D, and two relays A and B.Transmissions from S may be received by either or both of the relay nodes. . 35

4.7 The link capacity graph of the network in Figure 4.6. The dashed node rep-resents a virtual node, which accounts for packets that are received by bothof the relays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.8 A flow assignment that is compatible with the link capacity graph in Figure4.7 and achieves the maximum flow from S to D. . . . . . . . . . . . . . . . 37

v

4.9 A time-space representation of the network. Each row represents one physicalnode and each column represents a different time step. The amount of infor-mation transmitted from the source to the destination is the min-cut betweenthe top left node and the bottom right node. . . . . . . . . . . . . . . . . . . 37

4.10 A generalized Jackson network. The service times and routing decisions needonly be stationary and ergodic. . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 The transmission range R of a vehicular network is typically much larger thanthe width of the road. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 The connectivity of the VANET will be modeled by a line network, whereeach node connects to all neighbors within a distance R of itself. . . . . . . . 43

5.3 The connectivity of a VANET with evenly spaced nodes. Each vehicle hastwo neighbors in both the forward and reverse directions. . . . . . . . . . . . 44

5.4 A model of a vehicular network in which the source and destination can com-municate with their neighbors without randomly contending for the channel,as the relay nodes do. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 In a homogeneous vehicular network, all cuts that do not pass through theinfinite capacity edges near the source and destination will have the same(minimum) value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.6 The topology of the vehicular ad hoc network. The virtual nodes are repre-sented by the dashed circles, and the infinite capacity edges from the virtualnodes to the physical nodes by the bold lines. The minimum cut is a cutwith no infinite capacity edges passing through it from the source side to thedestination side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.7 To simplify computation of the min-cut of the vehicular network, the weightof the edge from A to C and the edge from A to the virtual node BC will becalculated in one step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.8 A numerical evaluation of the expression for the min-cut under slotted Aloha,for both the optimal transmission probability p and the nominal value p = 1

2K,

reveals that using the optimal value provides only a small improvement overthe nominal value, even for large K. . . . . . . . . . . . . . . . . . . . . . . . 48

5.9 A simulation of the end-to-end throughput as a function of the neighborhoodsize K for the homogeneous topology shown in Fig 5.4. The network codingis done over the real field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.10 A simulation of the end-to-end throughput as a function of the number ofnodes between the source and the destination for the homogeneous topologyshown in Fig 5.4. The neighborhood size is K = 5 and the coding is doneover the real field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.11 In unslotted Aloha, when a node transmits a packet (the shaded rectan-gle) neighboring nodes must be silent for two consecutive time slots (thecrosshatched rectangles) to prevent a collision. . . . . . . . . . . . . . . . . . 51

5.12 The optimal TDMA scheme is found by packing concurrent transmission astightly as possible without inducing collisions, and having each packet for-warded by the furthest node that received it. . . . . . . . . . . . . . . . . . . 51

vi

5.13 To a first order, a non-homogeneous vehicular network can be approximatedby a network with several different regions of constant connectivity. . . . . . 51

5.14 The number of independent packets received per epoch as a function of thelocation of the destination vehicle in the asynchronous transmission scheme.Results are shown for two strategies: routing with the asynchronous schemeand network coding with the asynchronous scheme. . . . . . . . . . . . . . . 54

5.15 The total number of received packets per epoch as a function of the locationof the destination vehicle in the asynchronous transmission scheme. This plotcounts all packets received without collision, regardless of whether they areindependent. Results are shown for two strategies: routing with the asyn-chronous scheme and network coding with the asynchronous scheme. . . . . 55

5.16 The time when the infostation begins transmitting each epoch, and the timewhen the destination vehicle receives the final packet of the epoch, for theasynchronous scheme. The vertical distance between the two curves is thetime delay of the asynchronous scheme with network coding. . . . . . . . . . 56

5.17 The number of independent packets received per wave as a function of thelocation of the destination vehicle. Results are shown for four strategies:routing with the synchronous overhearing scheme, network coding with thesynchronous overhearing scheme, routing with the asynchronous scheme, andnetwork coding with the asynchronous scheme. . . . . . . . . . . . . . . . . . 57

5.18 The time when the infostation begins transmitting each wave, and the timewhen the destination vehicle receives the final packet of the wave for networkcoding with the synchronous overhearing scheme and network coding with theasynchronous scheme. The vertical distance between the curves is the timedelay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

vii

List of Tables

3.1 The system parameters used to simulate the performance of the synchronoustransmission scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 The end-to-end throughput of the asynchronous transmission scheme, in thelimit of high density, for three common MAC protocols. . . . . . . . . . . . . 46

viii

Acknowledgments

I must first express my deep gratitude to my advisor, Prof. Kannan Ramchandran, forhis support, guidance, and unfailing encouragement all these years. This work would nothave been possible without his vision, assistance, and countless hours of research discussions.I deeply appreciate the fact that he never gave up on me.

I would like to thank Prof. Minghua Chen for his invaluable help during the past year. Icould not have successfully completed this research without his guidance and many valuableSkype discussions. I would also like to thank my committee members, Prof. Michael Gastparand Prof. Raja Sengupta, for their constructive and helpful feedback on my research and onmy dissertation.

I must thank all of my collaborators in graduate school. In particular, I must acknowledgeProf. Francois Baccelli for the time he spent discussing the connections between queueingtheory and wireless networks with me while in Berkeley, and Prof. Luca De Nardis for thetime he spent working with me when I was getting started on this project.

I also thank all of my fellow members of the BASiCS lab and the Wireless FoundationsCenter for all of the great memories from my years in graduate school.

Finally, I thank the Hertz Foundation for their financial support, which helped make itpossible for me to pursue the research presented in this dissertation.

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Chapter 1

Introduction

The design and deployment of safe and efficient transportation networks is one of themost important large-scale engineering challenges of the 21st century. In the US alone, thereare over 33,000 traffic fatalities per year, according to data from the National Highway SafetyAdministration [1]. Furthermore, the US Bureau of Transportation Statistics reports that in2007 drivers spent a total of 4.2 billion hours delayed in traffic [2]. Not only does this timerepresent a loss in productivity, but it also significantly increases fuel consumption and airpollution. In the near future these problems will likely increase in magnitude, as the numberof vehicles on the road worldwide is projected to double within the next decade [3].

One of the many strategies for making highways safer and less congested has been thedevelopment of intelligent transportation systems (ITS), the incorporation of informationand communication technologies in vehicles and transport infrastructure. There are a largeand growing number of systems, targeting a wide range of applications, that fall under theheading of ITS. For example, cooperative active safety systems enable numerous applica-tions that can mitigate collisions and traffic fatalities, such as collision warning, emergencyelectronic brake lights, and slow/stopped vehicle alerts [4, 5]. Two systems have recentlybeen developed which use smartphones in vehicles to estimate traffic conditions and traveltimes more accurately than by using existing roadside infrastructure [6, 7]. This traffic datacan then be relayed to drivers, allowing them to plan trips more effectively, and can alsoused by traffic management centers to plan long term upgrades to the highway system. Ina similar vein, the development of electronic tolling mechanisms will enable high occupancytoll lanes and congestion pricing, which can potentially reduce traffic congestion at peakhours [8]. The most forward looking ITS projects have even tested completely automatedvehicles, which drive themselves without input from a human [9]. Such automated highwaysystems have long been known to increase the capacity of roads by as much as four times[10].

The common technology underlying all of these disparate systems is wireless communi-cation. In each application, the vehicles on the road must be able to exchange informationeither with each other or with the fixed infrastructure units along the roadside. All realtime intelligent transportation systems require a wireless link which can provide a sufficientthroughput, delay, and reliability to each vehicle.

A variety of architectures have been proposed for providing wireless connectivity to vehi-

2

cles. One general approach is to extend legacy networks by placing devices in vehicles thatcan utilize existing infrastructure. Examples include using cellular networks to collect trafficdata generated by smartphones [11], and opportunistically making use of open WiFi accesspoints to deliver data to vehicles [12]. These networks are all based on a single hop topology,as shown in Figure 1.1. The vehicles use the wireless channel to communicate directly withone or more base stations, which may be linked by a backend wired network.

Figure 1.1: A single hop vehicular network. The vehicles communicate only with the fixedbase stations along the roadside.

A second general approach, in contrast, involves forming an ad hoc network amongthe vehicles on the road, as shown in Figure 1.2. In this paradigm vehicles communicatedirectly with each other, as opposed to communicating only with fixed access points [13, 14].Vehicular ad hoc networks (VANETs) are especially well suited for safety applications, wherelow latency is the most important system requirement. The development of VANETs wasfostered by the decision of the FCC to allocate 75 MHz of spectrum in the 5.9 GHz bandfor use by Dedicated Short Range Communication (DSRC) systems, which is intended to“improve traveler safety, decrease traffic congestion, facilitate the reduction of air pollution,and help to conserve vital fossil fuels” [15]. The availability of this band has promptedresearch and development of a variety of ITS services that could make use of it, targetingapplications beyond just vehicle safety.

In this dissertation, we consider the problem of designing communication protocols forvehicular ad hoc networks. In particular, we focus on the medium access control (MAC) andnetwork layers of the protocol stack. The fundamental characteristic which distinguishesVANETs from other wireless networks, even other ad hoc systems such as sensor networks,is the high mobility of the individual nodes. The network topology changes at a very highrate, due to vehicles entering and exiting the road as well as the fact that vehicles often travelat very different speeds. The result is that the coordination of transmissions and routing ofpackets is costly and difficult. In fact, vehicles may not even know their current neighbors.While neighbor discovery and self-organization in ad hoc networks are technically feasible

3

Figure 1.2: A vehicular ad hoc network. The vehicles communicate directly with each other,without requiring a base station to organize and manage the network.

in static scenarios [16, 17], they become much more challenging with high node mobility.Repeatedly executing such algorithms, each time the topology of the VANET changes, wouldquickly consume a large portion of the system resources.

We propose instead the use of simple uncoordinated protocols for VANETs. The key toolis to use randomization to combat the fact that no single vehicle or infrastructure element hasknowledge of the global network topology. Thus, each vehicle randomly and independentlyaccesses the channel, instead of attempting to schedule transmissions to avoid collisions andoptimally reuse the available bandwidth. The only assumption is that vehicles are timesynchronized, enabling the use of a slotted protocol. Similarly, instead of utilizing globalrouting tables, which require significant resources to initiate and maintain in a dynamicnetwork, each node randomly mixes the packets that it transmits. This protocol does notrequire any link level feedback as to which packets are received and which are lost due tocollisions.

We are primarily concerned with analyzing the performance provided by such uncoordi-nated protocols in vehicular ad hoc networks. We present a general method for computingthe throughput in networks using random channel access and random linear network coding,and then apply it to a model of the topology of a VANET. Our first main contribution showsthat on a congested highway, which can be modeled by a dense network with homogeneousconnectivity, the throughput is independent of the number of hops between the source anddestination. Thus, the network can provide a constant throughput over any distance, so longas there is the same density of vehicles. With a very simple distributed MAC protocol, thethroughput is about 45% of the throughput in a fully coordinated network. Thus, the “costof uncoordination” is a loss in performance of about a factor of 2. By using a more complexMAC protocol, such as by incorporating carrier sensing, this cost can be reduced.

The intuition behind this result is that network coding, the mixing of packets at interme-diate nodes, makes the uncoordinated network robust against packet losses. Without coding,in contrast, the throughput would degrade with distance due to the inability to coordinateamong nodes. In effect, each vehicle functions as a digital repeater.

In the course of studying the throughput of uncoordinated networks, we address theproblem of starvation, the possibility that intermediate nodes in a multihop network maytemporarily not have any new information because of the random ordering of transmissions.

4

We then use queueing theory to demonstrate our second main contribution, that starvationdoes not affect the end-to-end throughput. This result has application in a number of fieldsextending beyond vehicular networks. In particular, it is relevant to the study of CSMAnetworks, where it is often assumed for analytic tractability that all nodes are in a saturatedstate, meaning that they always have new information to transmit.

This dissertation is organized as follows. In Chapter 2, we give an explicit formulation ofthe multihop communication problem that we will study, and discuss the uncoordinated MACand network layer protocols that we will apply to this problem. In Chapter 3 we propose aspatially synchronized protocol for multihop communication in VANETs, which requires aminimum level of coordination among the vehicles to implement. The goal in this chapteris to explore the tradeoff between coordination and performance, by studying a protocolwith significant randomization but still a small degree of coordination. It also provides abenchmark, against which we can then compare a completely uncoordinated protocol. InChapter 4 we present a general method for computing the end-to-end throughput of multihopnetworks using an uncoordinated MAC protocol and network coding. Then in Chapter 5 weuse this algorithm to calculate the throughput of an uncoordinated VANET. We concludethe dissertation in Chapter 6 with a discussion of future work.

5

Chapter 2

Multihop Content Distribution in

Vehicular Ad Hoc Networks

In this dissertation, we focus on multihop content distribution in vehicular ad hoc net-works. There are two main features of this problem which distinguish it from other VANETcommunication problems. The first is that data must be transmitted over large distances,and thus a multihop communication protocol will be used. There are several scenarios wherethis is true. Clearly, one example is when the source and destination are separated by sucha large distance that single hop communication is not possible. A second example is whenlimitations on the transmit power make it necessary to relay data over multiple hops. Wecan imagine an emergent service being deployed in unlicensed bands with strict power limi-tations, as opposed to purchasing licensed spectrum where it might be possible to transmitdata directly from source to destination.

The second unique feature is that a relatively large amount of data must be sent, andtherefore it must be divided into numerous packets. One of the key results of this dissertationwill be to show that the data can be distributed over many hops without a loss in perfor-mance. There are a variety of applications where a large volume of data must be deliveredto vehicles on a road. As one example, we consider applications where the VANET is beingused to distribute data that is meaningful to drivers of vehicles. For example, the CaliforniaDepartment of Transportation has cameras placed near bridges and freeway interchanges inthe Bay Area [18]. We could use the vehicular network to distribute live video from suchcameras to vehicles, so drivers can make decisions about which road to take or whether todelay their trip. Similarly, we could distribute a real-time map showing the average speedson the roads in a metropolitan area, similar to the traffic map provided by Google maps.As a second general class of applications, we can consider the vehicular network to be abackbone which is used to distribute data that is independent of the fact that the network isformed on a highway. In the realm of entertainment, for example, we could use the multihopnetwork to stream a television show which passengers in the vehicles would watch duringtheir trip.

In order to situate this multihop content distribution problem within the larger fieldof vehicular networking, we briefly compare it to safety applications, which are one of thetechnologies driving the deployment of VANETs. In safety systems, the data is usually small

6

enough to be contained in a single packet, for example a warning that a vehicle is brakingsharply or changing lanes. In addition, safety messages are usually only relevant to vehiclesthat are near the source. For example, when a vehicle changes lanes, only the vehicles in theadjacent lane which are in the driver’s blind spot need be warned. Thus, safety messagesonly need to be transmitted over a small number of hops, if not just a single hop. Due tothese considerations, it suffices to flood the safety messages over the region where they arerelevant, and to use repetition coding to combat channel losses. For this class of applications,there is little benefit to using the more powerful coding strategies that are examined in thisdissertation.

The rest of this chapter is organized as follows. In Section 2.1 we formulate the multihopcommunication problem. In Sections 2.2 and 2.3, we discuss the MAC and network layerprotocols, respectively, that will be used in subsequent chapters to design a scheme formultihop VANET communication. In particular, in Section 2.3 we provide backgroundmaterial on network coding, which is the key component of our solution. In Section 2.4 wereview some related work on content distribution in VANETs.

2.1 Multihop Networks

We will begin by considering a unicast communication problem, where a fixed base stationwants to use the VANET to send data to a destination vehicle that is far down the road,as illustrated in Figure 2.1. We will also refer to this base station as an infostation, toemphasize that it is the source of the data, in contrast to cellular telephone networks whereone of the mobile devices is the source.

Figure 2.1: We consider a unicast communication problem in a multihop wireless vehicularad hoc network.

We make several simplifying assumptions. Primarily, we assume throughout this disser-tation that all vehicles use the same transmit power and thus have a common transmissionrange R meters. We also assume that the physical layer is modeled by the Gupta and Ku-mar protocol model for packet reception [19]. In that model, a vehicle can receive packetstransmitted by other vehicles that are within a distance R of itself, but cannot receive anypackets transmitted by vehicles further than a distance R from itself. If two vehicles withina distance R of the receiver transmit at the same time, the receiver hears a collision and

7

neither packet can be decoded. In the following chapters, we will describe various models ofthe spatial distribution of vehicles in more detail. For now, it suffices that we will assumethat there are no gaps longer than R between adjacent vehicles, and therefore there are nodisconnected components of the graph.

We also assume here that the vehicles and base stations use omnidirectional antennas. Amore advanced system could improve the throughput by using directional antennas. Evenplacing a relatively simple two antenna array at each vehicle could reduce interference byfocusing the radiated power in the direction of the destination. The analysis of a VANETwith directional antennas is left for future work. Finally, all packets in the network will havea fixed length.

As discussed in Chapter 1, the most important property of a VANET is that centralizedcoordination and control is either extremely expensive or technically infeasible. For thisreason, we will design distributed MAC and network layer protocols for this problem.

2.2 MAC Layer Protocols

The vehicles cannot coordinate when accessing the channel, due to the lack of knowledgeof the topology. Instead, we will utilize randomized MAC protocols, such as Aloha [20] andcarrier sense multiple access [21] based schemes.

2.2.1 Slotted Aloha

The slotted Aloha protocol was originally proposed for use in fully connected computernetworks [20]. The basic idea is that nodes access the common communication channel in-dependently of one another. Whenever a terminal has data to send, it transmits a packetin the next time slot. This protocol can be readily adapted to wireless vehicular networks.Time is partitioned into slots, and vehicles randomly choose to either transmit or receive ineach slot. Vehicle i will transmit a packet in every time slot with probability pi, independentof all other vehicles and all other time slots, and with probability (1− pi) will not transmit.Whenever a node does not transmit, it attempts to receive a packet. If exactly one neigh-boring vehicle within its transmission range is transmitting, then that packet is successfullydemodulated and decoded. If more than one neighbor transmits, the vehicle hears a collisionand is unable to decode any of the transmitted packets.

This protocol requires a very minor degree of coordination, because all vehicles musthave access to a common reference clock so that the time slot boundaries are synchronized.This could be implemented, for example, by placing a GPS receiver in each vehicle, whichgives them access to a universal time basis. Alternatively, a single high-power transmittercould broadcast beacons on an auxiliary channel. Since this transmitter is only demarcatingthe boundaries of the time slots, and not communicating information, only a very smallbandwidth would be required.

8

2.2.2 Unslotted Aloha

The unslotted Aloha protocol for computer networks is similar to slotted Aloha, exceptthat the terminals now transmit each packet immediately after it is generated, instead ofwaiting until the beginning of the next time slot. Thus, the transmission times are nowasynchronous. When two terminals transmit, the packets may partially overlap, which alsoresults in a collision and loss of information. In slotted Aloha, on the other hand, allnodes align their transmissions with the same slot boundaries, and therefore packets eithercompletely overlap in time or are received without collision.

In the context of wireless VANETs, we will use unslotted Aloha to refer to a protocolwhere each vehicle independently chooses to either transmit or receive in every time slot, butthe vehicles do not have access to a common reference clock. Each vehicle implements theslotted Aloha protocol described in the previous section, but the slot boundaries are shiftedby a random offset. In this case, transmitted packets can partially overlap, just as in theunslotted Aloha protocol from computer networking.

2.2.3 Carrier-Sense MAC

In Aloha-style protocols, many packets are corrupted by collisions, since neighboringnodes frequently transmit at the same time. The performance can be improved, withoutrequiring coordinated scheduling, by having each vehicle check whether the channel is in usebefore transmitting. When a node chooses to transmit, it will first sense the channel. If thechannel is free, it proceeds with its transmission. On the other hand, if the channel is busy,then the vehicle will back-off by a random amount of time before trying to transmit again.The design of carrier sense multiple access (CSMA) networks is a very active research area,with substantial work on optimizing the channel access and back-off parameters [22, 23, 24].

While carrier sensing can mitigate the losses inherent to Aloha protocols, collisions maystill occur due to the hidden node problem. This occurs when two transmitters do not senseeach other’s transmissions, but a receiver is within range of both of them. Thus, the twotransmitters may decide to transmit concurrently, which results in the receiver hearing acollision. However, recent work has shown that any network can be made hidden node-free,by adjusting the carrier sensing range to be large enough in proportion to the transmissionrange1 [25].

We also note that more advanced carrier-sense protocols often include collision avoidancemechanisms. In one standard scheme, a transmitter sends a request to send (RTS) packetbefore transmitting. If the receiver hears the RTS packet, it replies with a clear to send(CTS) packet. The transmitter only begins to transmit when it receives the CTS packet. Inthis dissertation, we do not consider the use of such collision avoidance protocols for severalreasons. First, our goal is to show that a significant throughput can be achieved with verysimple distributed protocols. Second, in Chapter 4 we will consider scenarios where vehicles

1This is true when there is a line of sight path between all pairs of nodes, and the hidden node problemoccurs due to propagation losses. If there are obstacles between some nodes, then the hidden node problemmay be unavoidable.

9

do not know the identity of their neighbors and cannot address each other, which makes itimpossible to use RTS/CTS packets.

2.3 Network Layer Protocols

The medium access layer only defines how the vehicles will access the channel. Thenetwork layer describes how information is moved from the source to destination. In thissection, we discuss several network layers that are compatible with the requirement that nosingle node has global knowledge of the network topology. This lack of coordination preventsthe computation and distribution of centralized routing tables

We first note that by identifying themselves to their neighbors, vehicles could learn thelocal topology of the network. By propagating this information throughout the network, itwould be possible in theory to learn the complete topology and then execute a predefinedalgorithm to construct common routing tables. However, such a scheme will consume asignificant amount of system resources, since the topology of the VANET changes frequentlydue to vehicles entering and exiting the road, and moving relative to each other. If vehiclesenter and exit the road on the same timescale as the algorithm that learns the topology,then the network will not be able to maintain accurate routing tables. We instead considerdistributed network layer protocols. By this, we refer to a general class of protocols wherevehicles select what data to include in a transmitted packet independently of each other. Inthis section we outline two such protocols, distributed routing and random network coding.

2.3.1 Distributed Routing

One possible network protocol is distributed store-and-forward routing. In this scheme,each vehicle maintains a buffer of all of the packets that it has received. When a vehicleaccesses the channel, it randomly selects one of the stored packets to transmit, independentlyof the other vehicles.

There are a variety of heuristics that could be used to select packets to transmit. Inthe simplest case, a vehicle would uniformly choose one of the packets stored in its buffer.A more advanced protocol could choose the more recently received packets with a higherprobability than those that have been in the buffer longer, since the newer packets are lesslikely to have been received by neighboring vehicles.

However, all variants of distributed routing suffer due to the lack of coordination. Becausevehicles do not know which packets are present at any other vehicles, and do not receivefeedback about which transmissions were received correctly and which were lost to collision,some transmissions are not useful. A transmitter may send a packet that its neighborsalready possess, when there are other packets in its buffer that its neighbors do not haveyet. The destination is essentially performing a coupon collecting operation, and may receivenumerous copies of one source packet before receiving any copies of another source packet.

10

2.3.2 Random Network Coding

A second general option is for the vehicles to use network coding. The fundamentalidea of network coding is to allow mixing of information at intermediate nodes, instead ofsimply routing it. The seminal paper by Ahlswede, et al. [26] proved that a network codewhich allows a source to send h units of data per unit time through the network to a set ofdestinations exists if and only if the maximum flow between the source and each destinationare all greater than or equal to h.

The maximum flow is the largest rate at which information can be delivered from asource to a destination, while respecting the capacities of each link. With network coding, itis possible for all of the destinations to receive data at a rate equal to the smallest maximumflow between the source and any destination, which is clearly the largest possible multicastrate. In general, this is not possible if the intermediate nodes perform only routing. Morerecently, it has been shown that the maximum rate can be achieved with high probabilityif each node simply transmits a random linear combination of its inputs (using coefficientsdrawn from a sufficiently large finite field) [27, 28]. Randomized network coding can bepractically performed without any centralized coordination [27], and is very useful in casesin which the topology of the network is unknown or dynamic [29].

In a vehicular network, the implementation of random linear network coding is particu-larly straightforward. Each packet in the network consists of simply a linear combination ofall of the source packets. For example, consider a scenario with three source packets labeledX1, X2, X3. Each Xi consists of a long vector of source symbols, which are elements in a finitefield. The packets transmitted by vehicles will be of the form Y = α1X1 + α2X2 + α3X3,where the mixing coefficients {αi, i = 1, 2, 3} are elements in the same finite field as thesource symbols. The actual packet will consist of a header, which contains the coefficientsαi, and the data, Y .

Figure 2.2: In a system based on random linear network coding, the intermediate nodestransmit linear combinations of the packets that they have received.

Every vehicle maintains a buffer containing all of the linearly independent packets thatit has received. When a vehicle chooses to transmit, it generates a set of random mixingcoefficients, uses those coefficients to combine the packets in its buffer, and transmits theresult over the channel. Figure 2.2 provides a simple example. The relay vehicle in themiddle has received two packets from its neighbors to the left, which represent the equationsY1 = 2X1 + 3X2 + X3 = 14 and Y2 = 3X1 + 5X2 − X3 = 23. While the coefficients in thisexample are integers, in actual implementations the coefficients will be elements in a finite

11

field. When the relay decides to transmit a packet, it randomly generates two local mixingcoefficients γ1 and γ2, and uses them to form the new equation

γ1Y1 + γ2Y2 = γ1(2X1 + 3X2 + X3) + γ2(3X1 + 5X2 − X3) = γ1(14) + γ2(23)

For example, if γ1 = −1 and γ2 = 2, then the new equation is 4X1 + 7X2 − 3X3 = 32, asshown in Figure 2.2. The header of the new packet must contain only the absolute coefficientvector (4, 7,−3). The local coefficients (γ1, γ2) do not need to be propagated through thenetwork.

Whenever a vehicle receives a new packet, it will check whether or not the packet isa linear combination of the packets already stored in the vehicle’s buffer. If the packet isa linear combination of packets that the vehicle has already received, then the vehicle candisregard the new packet. On the other hand, if the new packet is linearly independent of theset of prior packets, then the vehicle adds the new packet to the buffer. Once the number ofpackets in a vehicle’s buffer is equal to the number of source packets, the vehicle can recoverthose source packets by solving a system of equations.

Clearly, the overhead required to transmit the coefficients in each packet is proportionalto the number of source packets. In a streaming application, where there is essentially aninfinite amount of source data, the source packets are partitioned into blocks, referred to asepochs. Instead of mixing all source packets at the intermediate nodes, only packets thatare from the same epoch are combined with one another. Dividing the source data streaminto disjoint epochs keeps the overhead in each packet at a reasonable level.

Network coding provides two advantages over routing. First, nodes can simply broadcastcoded packets to all nodes within their transmission range, without needing to know the net-work topology. Second, a system using network coding can provide substantial throughputgains even when nodes do not receive any feedback as to which of their neighbors received atransmitted packet. This is important because the communication protocol can be substan-tially simplified if feedback is unnecessary. While providing feedback is not impossible in avehicular network, a protocol without feedback utilizes the scarce bandwidth resources moreefficiently. In addition, there are contention issues on the feedback channel as well. In thisdissertation, we analyze the end-to-end throughput when random linear coding is combinedwith the various distributed MAC protocols discussed in Section 2.2.

2.4 Related Work

The problem of distributing data from infostations to vehicles on a highway was firstexamined in [30, 31, 32]. This series of work focuses on a low power, low connectivitysetting. Vehicles are by default disconnected from the rest of the network, and exchangepackets over a short time period when they pass another vehicle in an adjacent lane. Thenumber of packets that can be received is a function of the relative speeds of the vehicles.Such a scenario is significantly different from the situation considered here, where a highdegree of connectivity is assumed at all times for all vehicles in the network.

In [33], the performance of network coding as a solution for content distribution in ve-hicular networks is investigated. The paper adopts a medium access protocol based on the

12

802.11(b) MAC, in combination with network coding. It analyzes, via computer simulations,the reduction in file download time as compared to a network layer based on Ad Hoc On De-mand Distance Vector (AODV) routing, originally proposed in [34]. The impact of mobilityand node density on the performance of network coding and routing is also briefly investi-gated, and simulation results indicate that network coding can take advantage of increasingmobility. It should be noted that these results were obtained in a scenario characterized byrandom car movement in an urban street grid, rather than on a highway.

Our work differs in several aspects. First, our goal is to understand the asymptotic per-formance limits for network coding in vehicular ad hoc networks, by providing a theoreticalanalysis and supporting it with computer simulations. In order to do so, we focus on an ana-lytically tractable scenario, constituted by a highway where cars adopt a slotted Aloha MACprotocol to cooperatively distribute content provided by an infostation, where the vehicledensity is relatively high.

Second, we carry out extensive throughput analysis in order to determine the capabilityof a peer-to-peer scheme using network coding to support real-time content streaming onhighways. This analysis is paramount to the aim of designing a content distribution systemfor highways, since it provides an indication of the amount of infrastructure and the associ-ated cost required for reliable communications. In this framework, we analyze the tradeoffbetween system complexity and performance, by comparing a spatially synchronized schemeusing position information with a fully uncoordinated scheme.

Our work was initially motivated by the problem of communication using untuned radiosin a dense sensor network, analyzed in [35]. In this work, a large number of small, inexpensivesensor nodes are used to relay information from a source to a destination. Because the radiosin these sensors use an on-chip resonator, instead of a more expensive quartz crystal, theircarrier frequencies are random. Because it is not known a priori which nodes can receivetransmissions from which other nodes, it is not possible to coordinate the dissemination ofinformation through the network.

13

Chapter 3

A Minimally Coordinated VANET

Communication Protocol

In this chapter we propose and analyze a minimally coordinated transmission protocolfor multihop content distribution. The basic idea underpinning this scheme is to partitionthe highway into spatial blocks, which are then used to facilitate the flow of information overmany hops. Only vehicles in certain spatial blocks will attempt to transmit in specific timewindows. However, the vehicles within a block will operate in an uncoordinated manner,randomly accessing the channel independently of each other. Therefore, this scheme requiressome degree of coordination to set up the spatial structure and coordinate transmissions atthe block level, as opposed to the purely uncoordinated protocols discussed in Chapter 2. Wewill refer to this protocol as a synchronous scheme, to emphasize the spatial synchronizationimposed by the block structure. Our objective in proposing this scheme is to demonstratethe performance that can be achieved with a small degree of coordination, and to provide abenchmark against which we can compare the uncoordinated protocol examined in Chapter5. The primary metric that we use to quantify the system performance is the end-to-endthroughput between the source and destination.

The rest of this chapter is organized as follows. In Section 3.1 we discuss the blockstructure and the minimally coordinated MAC protocol. In Section 3.2, we analyze thethroughput of this scheme with network coding. Then in Section 3.3 we discuss a modificationof the basic synchronous protocol which can improve performance. Finally in Section 3.4we give simulation results which compare the throughput of these two synchronous schemeswhen using both network coding and routing at the network layer.

3.1 Spatial Partitioning of the Road

The synchronous scheme is founded on dividing the highway into blocks, as shown inFigure 3.1. The length of each block is chosen to be 1

2R, where R is the transmission range

of the vehicles. By choosing 12R for the length of the blocks, it is guaranteed that under

the physical layer model discussed in Chapter 2, a transmission from any vehicle in a givenspatial block can be received by any vehicle in the adjacent block. Even when a vehicle at

14

the left edge of block i is transmitting, the vehicle all the way at the right edge of block i+1is within distance R of the transmitter and can decode the packet, so long as there is nointerference.

Figure 3.1: In the synchronous transmission scheme, the highway is partitioned into blocks,where the length of each block is half of the transmission range of the vehicles.

The synchronous communication protocol uses a variant of the slotted Aloha MAC pro-tocol discussed in Section 2.2. Aloha is modified so that vehicles in the same block randomlycompete with each other for the channel, but vehicles in adjacent blocks do not interferewith each other. To this end, we group the time slots into larger units, referred to as frames,that contain M slots each, as shown in Figure 3.2. M is a free parameter, entirely under thecontrol of the system designer.

Figure 3.2: In the synchronous transmission scheme, the time axis is divided into slots, whichare then grouped into frames composed of M slots.

At the macroscopic time scale of the frames, data is transmitted in a deterministic man-ner. In one frame of M time slots, the infostation will transmit M unique packets to thevehicles that are in block 1, i.e., the block which receives transmissions directly from theinfostation. In the next frame, the vehicles in block 1 will transmit the data which theyjust received to the vehicles in block 2. During each subsequent frame, this data will betransmitted to the next block on the highway, similar to a “bucket brigade”. We will referto these packets, which are forwarded from one spatial block to the next in each frame, asa wave of data. Figure 3.3 shows an example of the movement of waves among the blocks.The dark arrows denote that the vehicles in block 1 and block 5 are transmitting during thesame frame.

The blocks which are concurrently transmitting must be separated by sufficient distanceto ensure that there is no interference between these waves. It can easily be shown that theminimum separation is 4 blocks. To see that a smaller separation will not work, consider a

15

scenario where blocks 1 and 4 are transmitting in the same frame. A vehicle in block 2 couldbe within distance R of both a vehicle in block 1 and a vehicle in block 4. If the vehicles inblocks 1 and 4 transmit in the same slot, this vehicle in block 2 will hear a collision.

This receiver in block 2 is then unable to decode the transmission from block 1, whichcontains a packet in a new wave of data, because of interference from block 4, which containsold information that block 2 has already received. This interference between waves willdegrade the throughput, since vehicles in block 2 will receive fewer packets from block 1,and thus have fewer packets to forward to block 3.

On the other hand, if the waves are separated by 4 blocks, then no such interference willoccur. No vehicles in block 2 are within the transmission range of any of the vehicles inblock 5. Note that this does not mean that no collisions occur in the synchronous scheme.Clearly two vehicles in the same block might still transmit in the same time slot, resultingin collision. However, by separating the waves by 4 blocks, we have eliminated collisionscaused by transmissions in separate waves of data.

Figure 3.3: In the synchronous scheme, waves of data propagate along the highway, movingone block to the right in each frame. The dark arrows show two spatial blocks that aretransmitting during the same frame. The concurrent waves are sufficiently separated so thatthere is no interference between them.

At the time slot scale, however, the nodes do not schedule their transmissions. As wehave discussed earlier, it is not desirable, or even feasible, for centralized coordination to beimplemented in this network. In the frame in which block i transmits to block i + 1, everyvehicle in block i will randomly choose exactly one slot in the frame on which to transmit,uniformly and independently of all other vehicles. During that time slot, the vehicle willeither transmit one of the packets that it received from its adjacent block during the previousframe (when distributed routing is used) or a random linear combination of all of the packetsit received (when network coding is used). When two or more vehicles in block i choose thesame slot, a collision occurs and no vehicles in block i+1 can decode either packet. However,if exactly one vehicle in block i selects a given time slot, then all vehicles in block i + 1 willsuccessfully receive the transmitted packet.

This MAC protocol is a variation of the slotted Aloha protocol outlined in Chapter 2.The first modification is that vehicles only attempt to transmit during every fourth frame,instead of operating identically in every time slot. The second modification is that within theframe in which a given block is forwarding data, each vehicle transmits in exactly one of theM time slots in the frame, instead of making an independent random decision in every slot.

16

We note in passing that modifying the unslotted Aloha and carrier sense MAC protocols foruse in this synchronous scheme would be much more challenging, because of the requirementthat all transmissions from vehicles in the same block must be contained within a specificframe.

In order to implement the synchronous scheme, it is necessary that each vehicle knowsthe identity of the block in which it is located, as well as the boundaries of the time frames.Although this requires a certain level of coordination among the nodes, it is still feasible ina distributed network. If each vehicle contains a GPS receiver, the vehicles know their ownlocations with high precision, as well as the boundaries of the time slots. The additionalinformation required for the synchronous scheme can be provided by having the base stationperiodically transmit a wave that contains control information, including the infostationlocation and information on the frame length and frame boundaries, instead of data. Whena new vehicle enters the network, it will not transmit until it receives one such control wave.After receiving the control wave, however, the vehicle will be able to compute the locationof the block boundaries, and will be able to become a full participant in the network.

3.2 Throughput Of Synchronous Protocol

We analyze the throughput provided by this synchronous protocol when the vehicles usenetwork coding. In particular, we focus on the limit of a high vehicle density, i.e., a largenumber of vehicles in each spatial block. We first optimize M , the number of time slots ina frame, and then compute the end-to-end throughput.

3.2.1 Optimal Frame Length

The first question that we address is how to choose the optimal value of M , the numberof time slots in a frame. If M is too small, then it will be highly probable that two or morevehicles in the same block will transmit on any single slot and most packets will be lost tocollision. On the other hand, if M is large there will be few collisions, but many time slotswill be empty because vehicles transmit on only one slot per frame.

We will assume that the vehicles are equidistantly spaced on the road, with exactlyN vehicles in each block. Since we are primarily interested in quantifying the asymptoticthroughput when the vehicle density is large, this is a reasonable assumption. However, ifwe instead chose to analyze the low density case, it would be more correct to assume thatthe vehicles were distributed randomly, e.g. according to a Poisson point process. In thatcase, there would be a variable number of vehicles in each block.

Given that the vehicles are spaced equally, the system designer has some control over N ,because the length of each block in Figure 3.1 can be varied by adjusting the transmissionrange. However, due to limits on the maximum allowed transmission power in the frequencyband where the system is operating, the transmission range cannot be increased arbitrarily.Thus N is dependent to some degree on the physical traffic conditions on the road.

During the frame when the vehicles in a given block are transmitting, each of the Nvehicles randomly selects one of the M slots via a uniform distribution, independently of

17

all other vehicles in the block. Our goal is to maximize the total number of packets thatare successfully received by vehicles in the next block. This is equivalent to maximizingthe number of time slots on which exactly one vehicle transmits. By the symmetry in themedium access protocol, this is in turn equivalent to maximizing the probability that exactlyone vehicle transmits in any single time slot. We can express the probability that exactlyone of the N vehicles in a block selects a fixed slot as:

(

N1

)

p(1 − p)N−1 = N1

M

(

1 −1

M

)N−1

where p = 1/M is the probability that a given vehicle transmits during a fixed slot. We canoptimize this expression by taking the derivative with respect to M and setting the resultequal to zero.

N

[

−1

M2

(

1 −1

M

)N−1

+1

M(N − 1)

(

1 −1

M

)N−21

M2

]

= 0

N1

M2

(

1 −1

M

)N−2 [

−

(

1 −1

M

)

+N − 1

M

]

= 0

The optimum frame length is then found by solving the equation 1 − NM

= 0, which yieldsM = N . This should come as no surprise, because all of the nodes in a single block arewithin range of each other. It is well known that the throughput of a fully connected Alohanetwork is optimized when each of the N nodes transmits with probability 1

N, as shown in

[20].The probability that exactly one vehicle in the block transmits on a given slot, and thus

vehicles in the next block receive a packet without collision, is given by N 1N

(

1 − 1N

)N−1=

(

1 − 1N

)N−1. In the limit as N becomes large, this expression approaches 1

e. Also, given that

a vehicle transmits a packet, the probability that no other vehicles in the same block transmit

on the same slot is(

1 − 1N

)N−1→ 1

e. Each transmitted packet is successfully received with

probability 1e, and lost due to collision with probability 1 − 1

e.

3.2.2 Throughput with Network Coding

The second question that we address is the throughput of the multihop synchronoustransmission scheme. The fact that data is being funneled down the highway in waves facili-tates the computation of the throughput. If we can count the number of independent packetsthat the destination receives in a single wave, then finding the throughput is straightforward.If the destination receives an average of x independent packets per wave, then the through-put per wave is x

N, as each frame has a length of N slots. Because the waves are separated

by four blocks in order to prevent interference between waves, the true throughput of thesynchronous scheme will be x

4N. This throughput is measure in the abstract unit of “pack-

ets per time slot”. Multiplying by the packet length and dividing by the slot duration willproduce a throughput in the standard unit of bits per second.

18

To find the average number of independent packets that the destination receives in eachwave, we formulate a random graph which models the propagation of a wave of data throughthe network, as shown in Figure 3.4. In this graph, the N nodes in each column representthe N vehicles in each block on the highway1. The H columns represent H blocks betweenthe base station and the destination vehicle. Each node is connected to all of the nodes inthe previous column with a unit capacity link, since a vehicle could potentially receive onepacket from every vehicle in the previous block, in the case where all of those vehicles choseunique slots on which to transmit.

1

1

2

N

2 3 H

Figure 3.4: A random graph modeling the propagation of a wave of data through a vehic-ular ad hoc network using the synchronous transmission scheme. Each node, and all of itsincoming and outgoing edges, is independently deleted with probability 1 − 1/e.

Because a transmitted packet is corrupted by collision with probability 1 − 1/e, each ofthe vertices in the random graph is independently deleted with probability 1−1/e, in whichcase all of its incoming and outgoing links are also deleted2. The source of an edge that isnot deleted is uniformly distributed over the non-deleted vertices in the previous column,because given that exactly one vehicle transmits in a slot, the source of that transmission isequally likely to be any of the nodes that do not experience a collision. We will label thisgraph G1−1/e, to denote that the vertices are deleted with probability 1 − 1/e.

The maximum flow is the largest number of packets that can be transported from thesource to the destination of the graph under any protocol, while respecting the link capacities.In our synchronous scheme, each vehicle transmits a random linear combination of all of thepackets that it received from the previous block. However, with random network coding itis known that the maximum flow is achievable with high probability, as long as the codingis done over a sufficiently large finite field [29]. The challenge in this problem is to computethe maximum flow through the random graph in Figure 3.4.

Because all vehicles in the first block deterministically receive the M packets sent by theinfostation, that link is not the limiting bottleneck on the end-to-end flow. In addition, we

1The synchronous scheme can also accommodate situations where the number of vehicles in each blockis variable. There will be a small performance degradation, due to the fact that the fixed frame length cannot be optimal for every block.

2Here, we are making the approximation that each transmission is lost to collision independently, in orderto make the computation of the throughput tractable. This is valid in the limit of large N .

19

recall that in the synchronous scheme, all vehicles in the same block receive the same packets.In particular, the destination receives the same set of packets as all of the vehicles in thelast column. Therefore, the maximum flow between the source and destination is given bythe number of vertex-disjoint paths between the first and last columns of the random graphin Figure 3.4. The following theorem quantifies the achievable flows, in the limit of highdensity. The proof will be given after we prove two associated lemmas.

Theorem 1: For any constant β such that β < 1/e, the maximum flow through G1−1/e isgreater than βN with high probability as N goes to infinity, as long as H is subexponentialin N .

As we will see, the probability that this theorem does not hold decays to zero exponen-tially with N . We prove Theorem 1 via two lemmas. We use a modified version of a commontechnique in percolation theory, which relates the probability of having many disjoint end-to-end paths through a lattice to the probability of having a single end-to-end path. We willuse A to denote the event that there exists at least one end-to-end path from the first columnto the last column of the random graph. We will let Ar denote the event that the randomgraph has the property that the deletion of any r vertices results in a graph belonging to A.This is equivalent to saying that any graph in Ar has at least r+1 vertex-disjoint paths con-necting the first and last columns. As an example, Figure 3.5 shows an instantiation of therandom graph, where the black circles represent nodes that are present and the white circlesrepresent nodes that have been deleted. This graph has only one vertex-disjoint end-to-endpath, and an example path is shown. Figure 3.6, in contrast, shows another instantiationof the random graph, where the nodes are deleted with a lower probability. This graph hasr = 3 vertex-disjoint end-to-end paths. The following lemma relates the events A and Ar tothe probability of deleting vertices in the two graphs.

Lemma 1: Let r be a positive integer. Then

P (Gp2 /∈ Ar) ≤

(

q2

q2 − q1

)r

P (Gp1 /∈ A)

for each 0 ≤ p2 ≤ p1 ≤ 1. Here, q1 = 1 − p1 and q2 = 1 − p2, and Gp is the random graphwhere each vertex is independently deleted with probability p.

Proof of Lemma 1: The proof of this lemma is based on the proof in [36] of a related resultin percolation theory, and the adaptation of that proof to the problem of communicatingusing untuned radios in [35]. Let Xi,j ∀i ∈ {1, 2, ...N} and j ∈ {1, 2, ...H} be i.i.d. randomvariables, uniformly distributed in the interval [0, 1]. We assign the variable Xi,j to thevertex in row i and column j of a rectangular grid. We then create two random graphs Gp1

and Gp2, which have vertices deleted with probability p1 and p2 respectively, as follows. Wefirst insert edges from every vertex in columns 2 through H to all vertices in the previouscolumn. Then we generate Gp1 by deleting the vertex in row i, column j (and all incomingand outgoing edges) if and only if Xi,j ≤ p1. Gp2 is likewise generated by deleting each vertexif and only if Xi,j ≤ p2.

20

Figure 3.5: An instantiation of the random graph modeling the propagation of a single wavethrough the network. The white circles represent nodes that were deleted, and the blackcircles represent nodes that were not deleted. This graph has only a single vertex-disjointend-to-end path. One such path is outlined.

Figure 3.6: An instantiation of the random graph modeling the propagation of a single wavethrough the network. The white circles represent nodes that were deleted, and the blackcircles represent nodes that were not deleted. This graph has r = 3 vertex-disjoint end-to-endpaths. One set of three disjoint paths is outlined.

Our goal is to find a relationship between the probability that Gp2 ∈ Ar and the proba-bility that Gp1 ∈ A. We observe that if Gp2 /∈ Ar, then there must exist a set of vertices Tsuch that:

• None of the vertices in T are deleted in Gp2

• |T | < r

• The graph G∗p2

obtained by deleting the vertices in T from Gp2 satisfies G∗p2

/∈ A.

21

There may exist many such sets T . For our purposes, it is sufficient to consider anyone set with this property. Suppose that Gp2 /∈ Ar, and that every vertex i, j in the set Tsatisfies p2 < Xi,j < p1. It then follows from the third bullet point above that Gp1 /∈ A.After conditioning on a set T , the probability that p2 < Xi,j < p1 for all vertices in T isgiven by

[(p1 − p2)/(1 − p2)]|T | = [(q2 − q1)/q2]

|T |

Therefore

P (Gp1 /∈ A|Gp2 /∈ Ar) ≥

(

q2 − q1

q2

)r

Applying the definition of conditional probability shows that

P (Gp2 /∈ Ar) ≤

(

q2

q2 − q1

)r

P (Gp1 /∈ A ∩ Gp2 /∈ Ar)

Finally, the proof of Lemma 1 is completed by using the fact that

P (Gp1 /∈ A ∩ Gp2 /∈ Ar) ≤ P (Gp1 /∈ A) �

Lemma 1 can be used to prove Theorem 1 if we can show that the probability thatGp1 /∈ A decays exponentially to zero with N , for some p1. In other words, if we can showthat P (Gp1 /∈ A) ≤ e−αN , where α is a constant that depends on p1, then we can rewriteLemma 1 as

P (Gp2 /∈ Ar) ≤

(

q2

q2 − q1

)r

e−αN = er ln

“

q2q2−q1

”

−αN

and by choosing r = β ·N , it follows that the probability of not having (βN + 1) end-to-endedge disjoint paths is bounded by

P (Gp2 /∈ Ar) ≤ e−

“

α−β ln“

q2q2−q1

””

N

If β satisfies the condition

β < α/ ln

(

q2

q2 − q1

)

then the probability of not having βN end-to-end paths decays to zero exponentially. Anexponentially tight bound on P (Gp1 /∈ A) does in fact exist, as shown by Lemma 2.

Lemma 2: The probability that there are no end-to-end paths in Gp1 is bounded by

P (Gp1 /∈ A) ≤ HeN log p1

22

Proof of Lemma 2: Observe that if all of the nodes in a single column of the graph aredeleted, then Gp1 /∈ A. On the other hand, if at least one node in every column is notdeleted, then an end-to-end path exists, because every node in the graph is connected toall N nodes in the previous column. The probability that all nodes in a single column aredeleted is pN

1 , and, by the union bound, the probability that all N nodes in one or morecolumns are deleted is less than or equal to

HpN1 = HeN log p1 �

Proof of Theorem 1: We can rewrite the probability that Gp1 /∈ A as e−N(− log p1−(log H)/N).Combining Lemma 1 and Lemma 2, we see that the probability that there are not at leastβN paths through Gp2 decays exponentially if

β <− log p1 − (log H)/N

log(

1−p2

p1−p2

)

If H is constant, or growing subexponentially in N , then the (log H)/N term will go tozero in the limit of large N , and the expression becomes

β <− log p1

log(

1−p2

p1−p2

)

Remember that in the original vehicular network p2 = 1 − 1/e, the probability thatpackets are lost to collision when the frame length M is chosen to be equal to N . However,p1 is unrelated to the graph Gp2, so we are free to choose any value of p1, as long as it islarger than p2. We evaluate the bound on β above by setting p2 = 1 − 1/e and choosingp1 = 1 − ǫ, and then taking the limit as ǫ goes to 0.

− log(1 − ǫ)

log(

1/e1−ǫ−1+1/e

) =− log(1 − ǫ)

log(1/e) − log(1/e− ǫ)

=log(1 − ǫ)

1 + log(1/e − ǫ)

Applying L’Hopital’s rule, we find that as ǫ approaches 0, this bound approaches 1/e.This proves that, with high probability, the throughput of the random graph is at least N/e.�

Theorem 1 demonstrates that the maximum flow through the random graph is at leastN/e. We also observe that the maximum flow, on average, cannot exceed N/e. This canbe easily seen by looking at the first two blocks. With each of the N nodes in block 1randomly choosing one of the N time slots, the vehicles in block 2 will receive, on average,N/e packets. Furthermore, nodes in subsequent blocks cannot receive more packets than

23

the nodes in block 2. Thus, we can make the stronger statement that the maximum flowthrough the graph is equal to N/e.

The implication is that a fraction (1 − 1/e) of the packets are lost between the first twoblocks, but with high probability no more information is lost in any subsequent hops (aslong as H does not grow exponentially in N). Each block in the system can be viewed as adigital repeater, which preserves all information that it received from the preceding block.

As discussed earlier, because the maximum flow for a single wave is N/e packets, andnetwork coding can achieve the maximum flow with high probability, the throughput of thesynchronous scheme is equal to 1/(4e) ≈ 0.09 packets per time slot. Thus, the destinationvehicle receives a novel packet in approximately 9 percent of the time slots.

Finally, it is instructive to compare this synchronous scheme with a fully coordinatednetwork. If complete coordination is possible, then each node in a block can be assigned aunique time slot within the frame, so that no collisions occur. In this case, the destinationreceives N independent packets per wave, and the corresponding throughput is 1/4 packetsper time slot. Hence, the “cost of uncoordination” in the synchronous scheme is a factor of1/e. This protocol achieves a throughput that is about 36% of what could be achieved witha fully coordinated block synchronized protocol.

Despite this loss, this scheme still enables us to transmit the same throughput over alarge distance, as long as the number of hops does not grow exponentially in the density.Like a fully coordinated protocol, the synchronous scheme has the attractive property of notlosing information as the number of hops grows, even though it involves random channelaccess.

3.3 Synchronous Transmission with Overhearing

While the block structure described in Section 3.1 is very helpful in moving waves ofdata along the highway in an analytically tractable manner, it has a shortcoming in that itartificially limits the number of packets that each vehicle receives. As one example, becausethe length of each block is R/2, when a vehicle in block 1 in Figure 3.1 transmits, somevehicles in block 3 will be able to successfully receive the packet. A second example is thatif two vehicles in block 1 transmit in the same time slot, all vehicles in block 2 will hear acollision, but some vehicles in block 3 will be able to receive one of the packets, by virtue ofbeing within range of one of the transmitters but not the other. Likewise, if vehicles listento transmissions from other vehicles within their own block, they might also receive newinformation3.

Taken together, these examples raise the possibility that a vehicle could improve thenumber of independent packets that it receives in each wave of data by decoding the trans-missions from all nodes within its range, instead of just transmissions from vehicles in the

3In the basic synchronous scheme, all vehicles in one block receive the same set of packets from theprevious block, and thus decoding transmissions from neighbors in the same block provides no additionalinformation. However, when vehicles are allowed to receive packets transmitted from non-adjacent blocks,it is no longer true that all vehicles in a block must have the same information. In that case, listening totransmission from a vehicle’s own block may be beneficial.

24

previous block. While many of these additional received packets will clearly provide re-dundant information, some of them could be independent of the packets received from theprevious block. We use the term “overhearing” to describe this extension of the basic syn-chronous scheme. In general, there is no reason not to receive and decode any transmissionnot corrupted by collision, since battery life is not a constraint in the vehicular network-ing application. The inclusion of overhearing complicates the analysis, and thus a rigorousbound similar to Theorem 1 for the basic synchronous scheme is beyond the scope of thiswork. However, in the next section we provide simulations that demonstrate the gains thatare achievable when overhearing is included.

3.4 Simulation Results

The synchronous transmission scheme was simulated in the OMNeT framework [37],under the following scenario. We considered a highway strip of length D, populated by Ncars

evenly spaced cars. The highway was served by an infostation, located along the strip inposition xIS = R/4, where R is the transmission range, assumed to be identical for theinfostation and the vehicles.

The transmission scheme used in the simulations can be summarized as follows.

1. The cars on the highway strip are partitioned into blocks of length R/2.

2. Time is divided into slots of duration Ts, which are in turn organized into frames of Nslots, leading to a frame duration Tf = N · Ts. The duration of a slot is selected as thetime required to send a packet of length Lpacket at a bit rate W .

3. Both the infostation and each block of cars transmit in one out of every four frames.

4. During frames in which transmission for a block is allowed, every car in the blocktransmits in each slot with probability p = 1

N.

We simulated Nwaves waves of information and averaged the measured throughput, cor-responding to a total simulation time of Tsim = 4 · Tf · Nwaves.

The values used for the various parameters of the network and the synchronous protocolare given in Table 3.1. We can make several observations about the scenario that is sim-ulated here. First, the duration of the simulation time Tsim, although sufficient to analyzethe behavior of the system over several waves, is short enough to allow us to neglect thedisplacement of vehicles with respect to the infostation caused by their movement at speedvcars. This justifies the assumption of a static scenario adopted in our analysis. Second, thevehicle density is fairly high, corresponding to a traffic jam scenario.

The above scheme was used to compare the performance of four different strategies,which are given by combining the two network layer protocols and the two variations onwhen vehicles listen to the channel:

• Routing with the basic synchronous protocol

• Routing with overhearing

25

Table 3.1: The system parameters used to simulate the performance of the synchronoustransmission scheme

Parameter Value

D 8000 mNcars 2400

R 800 mTs 1 ms

Lpacket 976 bitW 1 Mb/sN 120Tf 0.12 sp 0.0083

Nwaves 20Tsim 9.6 s

• Network coding with the basic synchronous protocol

• Network coding with overhearing

For each strategy we measured the system rank received per wave, averaged over Nwaves

waves. System rank was defined as the number of unique packets in the case of routing, andas the number of linearly independent packets in the case of network coding.

Figure 3.7 confirms the analysis in Section 3.2. This plot shows the system rank as afunction of the distance of the car from the infostation. For clarity, only the rank for the carat the center of each block is shown. These simulation results demonstrate that the use ofnetwork coding in place of routing significantly increases performance. More importantly, thesimulation confirms that network coding leads to a different scaling law than routing. Evenin the case of overhearing, in fact, the rank clearly decreases with distance under routing,eventually decaying to zero. On the other hand, network coding is able to support a highrank over large distances, as was proven analytically, and thus provides an appealing solutionfor streaming data delivery between roadside infostations. Also note that the introductionof overhearing is seen to improve throughput by about a factor of 1.75.

As an aside, we also verify via simulation that the increase in performance with networkcoding is independent of the number of delivered packets. Figure 3.8 shows the number ofpackets which are successfully received per wave, regardless of whether or not those packetsare independent. Because we selected the duration of the frame to maximize the number ofcorrect receptions, the number of received packets per wave is very close to the expected valueof N/e in the case of the basic synchronous scheme, without overhearing. The importantpoint is that the same number of raw packets is received with both routing and networkcoding. With overhearing, the total number of received packets per wave increases to 4N/e.However, many more of these packets will be dependent, and thus the throughput does notincrease by a factor of four with overhearing.

26

Finally, we also empirically studied the delay introduced by synchronous strategies. Fig-ure 3.9 shows the time when the infostation begins transmitting the first packet of each wave,as well as the time when the destination node receives the last packet of that wave. Theresults are shown only for network coding with overhearing. In this synchronous scheme,the delay with routing would be the same, although the destination would receive fewerindependent packets in each wave.

The figure verifies that the delay experienced between one wave and the next is constant,and equal to the delay introduced by the infostation, which transmits a new wave in everyfourth frame. The implication is that the synchronous scheme is suitable for deliveringdelay-sensitive traffic, such as streaming audio and video. It is also evident from Figure3.9 that synchronous strategies introduce a significant latency before the first wave reachesthe destination, due to the pipeline-like nature of the scheme. It can be readily shown thatthe delay for each wave is about (Nblocks + 2) · Tf seconds. For the settings used in thesesimulations, this corresponds exactly to the 22 · 0.12 = 2.64 seconds shown in Figure 3.9.The conclusion is that the synchronous scheme may not be suitable for use in interactiveapplications, where latency must be minimized.

0 1000 2000 3000 4000 5000 6000 7000 80000

20

40

60

80

100

120

Distance (m)

Ran

k pe

r w

ave

Synchronous RoutingSynchronous Routing + OverhearingSynchronous Network CodingSynchronous Network Coding + Overhearing

Figure 3.7: The number of independent packets received per wave as a function of the locationof the destination vehicle in the synchronous transmission scheme. Results are shown for fourdifferent strategies: routing with the basic synchronous protocol, routing with overhearing,network coding with the basic synchronous protocol, and network coding with overhearing.

27

0 1000 2000 3000 4000 5000 6000 7000 80000

50

100

150

200

250

Distance (m)

Rec

eive

d pa

cket

s pe

r w

ave

Synchronous RoutingSynchronous Routing + OverhearingSynchronous Network CodingSynchronous Network Coding + Overhearing

Figure 3.8: The total number of received packets per wave as a function of the locationof the destination vehicle in the synchronous transmission scheme. This plot counts allpackets received without collision, regardless of whether they are independent. Results areshown for four different strategies: routing with the basic synchronous protocol, routing withoverhearing, network coding with the basic synchronous protocol, and network coding withoverhearing.

28

1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

Wave

Tim

e (s

)

Start of TransmissionCompletion Time for Synchronous Network Coding + Overhearing

Figure 3.9: The time when the infostation begins transmitting each wave, and the time whenthe destination vehicle receives the final packet of the wave, for the scenario outlined inTable 3.1. The vertical distance between the two curves is the time delay of the synchronousscheme.

29

Chapter 4

Uncoordinated Wireless Networks

We now examine the throughput of fully uncoordinated networks, which do not requirethe minimal level of coordination necessary for the synchronous scheme considered in Chapter3. In this chapter, we will give a general method for computing the throughput of a wirelessnetwork with arbitrary topology that utilizes a randomized MAC protocol and random linearnetwork coding. Then in Chapter 5 we will apply this algorithm to the topology of a vehicularad hoc network and compute the throughput of an uncoordinated VANET.

In this chapter, we consider an even stronger restriction than our definition of unco-ordinated networks in Chapter 2. Previously, we had simply disallowed any centralizedscheduling of transmissions or routing of packets. By this definition, an uncoordinated net-work could still incorporate link level feedback and could potentially build routing tables vialocal interaction, although for simplicity in Chapter 3 we designed a scheme that did notrequire feedback.

In this chapter, in contrast, we consider the throughput of neighbor-oblivious wirelessnetworks. This refers to networks where nodes do not know the identities of any of theirneighbors. Thus, they cannot address neighboring nodes, and sending feedback about suc-cessfully received packets is not possible. This is a stricter limitation, which forces thenetwork to use distributed MAC and network layer protocols. Clearly, a protocol whichsatisfies the requirements of a neighbor-oblivious network can be used in an uncoordinatedVANET.

The rest of this chapter is organized as follows. In Section 4.1 we discuss the relationshipbetween this neighbor-oblivious problem and prior work on scheduling in wireless networks.In Section 4.2 we review the protocols that we use, and discuss the primary challenge incalculating the multihop throughput. Then in Section 4.3 we present a method for computingthe capacity of an uncoordinated network with arbitrary topology.

4.1 Neighbor-Oblivious Networks

One of the most challenging problems in wireless networking has proven to be scheduling:the question of how the nodes should share the limited wireless resource in order to max-imize performance. A throughput-optimal centralized scheme, known as maximal-weight

30

scheduling, was first presented in [38]. Jiang and Walrand made a major contribution tothe field by developing a distributed algorithm that uses carrier sensing to attain the op-timal throughput [22]. A similar distributed algorithm has been proposed for wavelengthassignment in optical networks [39]. Subsequent work has extended [22] to the discrete-timesetting [24, 23], characterized the delay [23], and implemented the algorithm in practice [40].

However, these schemes all make the implicit assumption that once the resource allocationproblem is solved, packets can be routed from the source to the destination, either via aglobal routing table or by implementing a distributed routing algorithm. Nodes must knowthe identities of their neighbors in order to forward packets to the next hop in the desiredpath. In this chapter, we consider multihop communication in neighbor-oblivious wirelessnetworks. When nodes do not know who their neighbors are, they cannot address packets tospecific neighbors and thus it is not possible to route packets. One obvious, though limited,approach is to use a simple packet flooding scheme. We present a more complex scheme thatcombines the random medium access control protocols outlined in Chapter 2 with networkcoding to achieve a higher throughput. Our focus will be on providing a method to computethe end-to-end capacity.

As in Chapter 2, we consider a unicast communication problem. We study the end-to-end throughput in a multihop network with arbitrary topology, as shown in Figure 4.1.Furthermore, we require that the nodes must be neighbor-oblivious, i.e., that they do notknow the identities of neighboring nodes and thus cannot route packets. Instead, when nodestransmit they must broadcast packets to all of their neighbors.

S

D

Figure 4.1: We consider a unicast communication problem in a multihop neighbor-obliviousnetwork. Our objective is to find the maximum throughput under the limitation that noneof the nodes know the identities of their neighbors.

4.2 A Distributed Communication Protocol for Neighbor-

Oblivious Networks

Due to the constraint that nodes do not know the identity of their neighbors, all nodesmust use one of the distributed MAC protocols discussed in Chapter 2 and a distributednetwork layer protocol. In this chapter, we analyze the throughput under the combinationof random network coding and a randomized MAC.

31

The primary challenge when calculating the multihop throughput is to account for thepossibility that when an intermediate node accesses the channel, it might not have any newinformation to send to its neighbors. Even if the transmitted packet is received withoutcollision, it does not convey any useful information to the receivers and the time slot is effec-tively wasted. This problem can be best illustrated through a concrete example. Considerthe network in Figure 4.2, where a source S sends packets to a destination D through asingle relay node R. Further, imagine that due to the random nature of the medium access,the source and relay nodes happen to access the channel in four consecutive time slots inthe order S, R, R, S.

DRS

Figure 4.2: This simple network illustrates the starvation problem that can occur when nodesaccess the channel in a random order. Consider a case when the source and relay transmitin the order S, R, R. When the relay transmits the second time, it has not yet received asecond packet from the source and hence can only send a redundant packet to D.

Even though S and R both transmitted two packets, D has only received one independentpacket. This is because when R transmits the second time, it has not yet received anotherpacket from S, and hence must retransmit the same packet that it sent previously. Althoughboth links may support the same average rate, the relay temporarily did not have any newpackets to send to the destination. We refer to this as the starvation problem to emphasizethat the relay has “consumed” all of its packets (forwarded them to the destination) withoutbeing “fed” (received new packets from the source).

We emphasize that this is not a problem that can be solved by coding. Even if R isusing network coding, when it transmits the second time it can only send a multiple of thefirst packet, which does not provide an independent packet to D. This starvation problem iscaused by the random ordering of when links are used, and it can occur regardless of whetherthe network uses routing or coding at the network layer. The source of the problem is thefact that the randomized MAC induces a random ordering on transmissions, as opposed toa deterministic ordering which could be implemented with centralized scheduling.

In wired networks, starvation is not an issue because all of the links in the network can beused concurrently. In a wired network with the same topology as Figure 4.2, both S and Rcan transmit packets in every time slot. Hence, R always has a new packet, which it receivedin the previous slot, to send to D. Starvation is also not an issue in the spatially synchronizedprotocol studied in Chapter 3. Because of the manner in which waves propagated from blockto block down the road in the synchronous protocol, each vehicle only transmitted a singlepacket in each wave. When this transmission occurs, the vehicle always has received someset of packets from the previous block, and hence transmits one linear combination of thosepackets. However, in more general networks where a node can transmit multiple packetscontaining combinations of the same data, we must account for starvation.

32

Much of the existing work on designing and analyzing random distributed schedulingalgorithms makes the assumption that nodes are in a saturated state, i.e., that they alwayshave a useful packet to send when they access the channel [22, 24]. This is equivalent toassuming that the starvation effect can be ignored. In the course of analyzing uncoordinatednetworks, we show that while starvation does lead to some transmissions being wasted, theaverage end-to-end throughput is not decreased. In essence, the loss due to starvation is asub-linear effect. Our results in this chapter validate the standard saturation assumption,and thus are useful in a wide class of random scheduling problems beyond the neighbor-oblivious networks considered here.

4.3 Capacity of Neighbor-Oblivious Networks

We now present an algorithm for computing the end-to-end throughput of a wirelessnetwork with arbitrary topology that uses a randomized MAC protocol and random networkcoding. The method can be summarized as follows. We will discuss the various steps in moredetail in the following sections.

1. Enumerate the independent sets, the sets of links in the network that can be usedconcurrently.

2. Compute the capacities of the individual links from a Markov chain on the independentsets.

3. Construct an equivalent wired network, with the same topology and link capacities asthe wireless network.

4. Find the minimum cut of the equivalent wired network.

The minimum cut of the equivalent wired network is clearly an upper bound on thethroughput of the wireless network, since the nodes in a wired network are always saturateddue to the fact that all links can be used concurrently. We prove that in fact this minimumcut can also be achieved in the wireless network. The key tool is the construction of aqueueing network which models the flow of equations in the wireless network. We usequeueing theory to show that whenever the source in the wireless network transmits dataat a rate smaller than the min-cut of the equivalent wired network, the destination receivesindependent packets at the same rate. Therefore, the maximum end-to-end throughput ofthe wireless network is the same as in the wired network, despite the fact that the starvationproblem may affect some transmissions.

4.3.1 Computation of Link Capacities

Our algorithm incorporates a recently proposed method for finding the capacities ofindividual links in CSMA networks. The throughput of links was shown to be computablefrom a time-reversible Markov chain in [41]. An efficiently computable approximation wasgiven in [42] and the approach has been further developed in [43].

33

As a concrete example, we will consider the linear network shown in Figure 4.3, with linkslabeled 1 through 4. The first step is to list the sets of links that can be used concurrentlywithout interfering with each other, which are known as independent sets. Trivially, any setcontaining only one of the links is an independent set, because it is always possible to useonly one link at a time. In addition, we note that links 1 and 4 can be used at the same timewithout interfering, and thus {1, 4} is an independent set. However, no other set of multiplelinks can be used concurrently. For example, imagine that node S attempts to transmit apacket on link 1 and node B attempts to transmit a packet on link 3 at the same time.Because A is within transmission range of both S and B, a collision occurs and the packeton link 1 cannot be received. Thus, there are five independent sets in this example: {1},{2}, {3}, {4}, and {1, 4}.

S DA B C1 2 3 4

Figure 4.3: A linear wireless network with four links. The independent sets, which are setsof links that can be used concurrently without interfering with each other, are {1}, {2}, {3},{4}, and {1, 4}.

The second step is to form a Markov chain whose current state is the set of links that arecarrying information at the current time. We refer to this set as the active links. Clearly,the state space of the Markov chain is given by the independent sets plus the null set.This Markov analysis is valid for networks using Aloha and CSMA medium access protocolswithout hidden nodes1.

The valid transitions of this Markov chain, and the transition probabilities, depend on thespecific MAC protocol being used. For slotted Aloha, the Markov chain is a fully connectedgraph, since the set of active links in two different time slots is independent. The networkcan transition from any independent set to any other independent set in a single time step.With CSMA, on the other hand, time dependencies are introduced because of the backoffcounter that is started when a node senses that the channel is in use. The transition diagramfor the network in Figure 4.3 will have a form like that shown in Figure 4.4.

Next, we find the steady state distribution of this Markov chain. Whenever there is afinite number of links in the network, there will be a finite number of independent sets.The Markov chain then has a steady state distribution, which can be computed from thetransition probabilities. Finally, the link capacities are given by the fraction of time when alink is active, which is the sum of the steady state probabilities of independent sets containingthat link. In this example, C1, the capacity of link 1, is equal to π1 + π14.

We can then create an equivalent wired network, which has the same topology and linkcapacities as the wireless network. Figure 4.5 shows this equivalent wired network for theexample in Figure 4.3. We will also refer to this graph as the link capacity graph.

1As discussed in Chapter 2, any CSMA network can be made free of hidden nodes by appropriatelyadjusting the carrier sensing range.

34

1

2

3

4

1,4∅

Figure 4.4: The set of links that are currently active can be modeled with a Markov chaindefined on the independent sets. This figure shows the Markov chain corresponding to thenetwork in Figure 4.3 when using a carrier sensing MAC protocol.

S DA B CC1 C2 C3 C4

Figure 4.5: The wireless network in Figure 4.3 can be replaced with an equivalent wirednetwork, know as a link capacity graph, whose individual links have the same capacities asthe wireless links.

A simple application of the max-flow min-cut theorem shows that the capacity of theequivalent wired network is equal to the minimum cut of this link capacity graph. Further-more, as stated in Chapter 3, random network coding can achieve this min-cut if the codingis done over a large enough field. However, it is not obvious that the capacity of the wirelessnetwork is the same as the equivalent wired network, due to the possibility of starvation atthe intermediate nodes.

4.3.2 Virtual Relay Nodes

Before addressing the relationship between the wireless and wired networks, we take abrief detour to introduce a more general example network. In the simple linear networkin Figure 4.3, each transmitted packet is received by only a single receiver. This masksthe broadcast nature of the wireless channel, where a single transmission could be receivedby multiple nodes. We now consider the example shown in Figure 4.6. When S transmits,depending on which of the extraneous interferers are simultaneously transmitting, the packetcould be received by A only, by B only, or by both A and B. However, in the last case, Aand B receive the same packet. It is not possible for S to send independent packets to Aand B in a single time slot, even though there are links to both A and B.

35

ℓ3

S

A

B

Dℓ1

ℓ2

ℓ4

Figure 4.6: A diamond network with source S, destination D, and two relays A and B.Transmissions from S may be received by either or both of the relay nodes.

To account for this effect, we introduce a virtual node AB, which represents packets thatare received by both of the relay nodes. The link capacity graph for this example is shownin Figure 4.7. Note that there are infinite capacity links from the virtual node to both ofthe relays, since packets sent to the virtual node are physically received by both A and B.

In the remainder of this section, we will use this example, since it brings out the generalstructure that will be needed when we study vehicular networks in Chapter 5

4.3.3 Achievability of Min-Cut of Equivalent Wired Network

Our main result is that the end-to-end capacity of the wireless network is equal to theminimum cut of the equivalent wired network, even when intermediate nodes experiencestarvation. A natural tool to track the propagation of information through a multihop net-work is queueing theory. However, there is an immediate complication in neighbor-obliviouswireless networks, where nodes are required to broadcast their transmissions. If a transmit-ted packet is received by two or more nodes, all of the receivers will store a copy of it intheir buffer. In queueing theory, in contrast, it is generally assumed that when a customeris processed by one server, it is then forwarded to the queue of only one other server.

In order to apply queueing theoretic results, we first introduce a genie into the systemwhich allows only one receiver to store each transmitted packet, even if multiple nodeswere able to receive the packet. We prove that this genie can only decrease the end-to-endthroughput of the network. Therefore, analyzing the system with the genie provides a lowerbound on the performance of the true system. Finally, we show that there is one geniesuch that the end-to-end throughput is given by the minimum cut of the equivalent wirednetwork.

36

B

C1

C2

C3

∞

∞

S AB

A

DC5

C4

Figure 4.7: The link capacity graph of the network in Figure 4.6. The dashed node representsa virtual node, which accounts for packets that are received by both of the relays.

Genie-Degraded Wireless Network

We first find the maximum flow through the equivalent wired network, and one rateallocation that achieves this flow. We denote the rate assigned to each link by R∗

i , and thevalue of the flow by λ∗. By the max-flow min-cut theorem, λ∗ is equal to the min-cut ofthe graph. Clearly, if the underlying physical network was a wired network with the samecapacities, then λ∗ would be the end-to-end capacity.

As an example, Figure 4.8 shows this rate allocation graph corresponding to the linkcapacity graph in Figure 4.7. Because this is a valid flow, R∗

i < Ci for all i, and the totalrate entering each intermediate node equals the total rate exiting. We further note that forany λ < λ∗, we can write λ = γλ∗, where 0 < γ < 1. Then, Ri = γR∗

i is a valid flow thatachieves rate λ.

To find the end-to-end capacity of the wireless network, we would like to construct anetwork of queues with the same topology as the wireless network. However, as notedabove, in a neighbor-oblivious wireless network, nodes broadcast all transmissions. We thusintroduce a genie, which alters the physical network in two ways. First, each node randomlydrops incoming packets with probability 1− Ri

Ci. Thus, the rate of each link is now Ri, instead

of the link capacity Ci. Second, packets entering a virtual node are assigned to exactly onephysical node, in a ratio proportional to the outgoing rates from the virtual node. Thus,there is now only a single copy of each transmitted packet, regardless of how many nodesreceive it, and standard queueing methods can be applied.

We note that in both of these modifications, the action of the genie is to force some nodesto drop packets that they actually received. We prove in Lemma 3 below that this geniecan only decrease the end-to-end throughput. Thus, the genie-degraded network provides alower bound on the actual system.

Lemma 3: The unicast capacity of a network using random network coding cannot increase

37

B

S AB

A

Dλ∗

R∗1

R∗2

R∗3

R∗4

R∗6

R∗7

R∗5

Figure 4.8: A flow assignment that is compatible with the link capacity graph in Figure 4.7and achieves the maximum flow from S to D.

when nodes discard a subset of packets that were successfully received.

Proof of Lemma 3: We begin by examining a time-space representation of the system,shown in Figure 4.9. All of the nodes in a row represent the same physical node, witheach column corresponding to a separate time step. Thus, we can think of the graph as“unwrapping” the time axis. We are interested in the max-flow from the source at the firsttime step (the top left node) to the destination at the final time step (the bottom rightnode). Observe that the horizontal edges represent each node’s internal buffer, i.e., eachphysical node stores all packets that it receives and can code over them in future time slots.The diagonal edges represent successful transmissions between nodes.

S

D

Figure 4.9: A time-space representation of the network. Each row represents one physicalnode and each column represents a different time step. The amount of information trans-mitted from the source to the destination is the min-cut between the top left node and thebottom right node.

A node ignoring a packet that it received is equivalent to removing one of the diagonaledges in the time-space graph, which cannot increase the min-cut. Thus, when coding overan infinite field, where the max-flow always equals the min-cut, the effect of removing some

38

edges is that the max-flow either is unchanged or decreases.When coding over a finite field, the probability that the destination can decode the source

data is at least (1 − 1/q)η where q is the field size and η is the number of links carryingcoded packets [29]. Removing edges will reduce η, which increases this lower bound on theprobability of decoding. Thus, removing links has not increased the min-cut, but we mustincrease the field size q if we want to guarantee that the decoding probability in the originalsystem meets a desired target. �

With the genie inserted, there is now only one copy of each packet in the network, evenif a transmission was physically received by multiple nodes. Therefore, we can represent theflow of information in the genie-degraded system by a queueing network. The queue lengthat each node represents how many more packets a node has received than it has successfullytransmitted to one of its neighbors2.

We also note that because of interference constraints in the underlying wireless network,it is not possible for adjacent servers in the queueing network to process packets at thesame time. Therefore, the server processes are correlated across the nodes, and the queueingnetwork is a generalized Jackson network, as opposed to a standard Jackson network withindependent service processes.

Stability Conditions of Generalized Jackson Networks

A generalized Jackson network refers to a network of queues where the only requirementson the arrivals, service times, and routing decisions are that they are stationary and ergodic,i.e., the service times at different servers do not have to be independent. An example networkis shown in Figure 4.10. A fundamental problem is to find the conditions on the arrival rateλ such that the queueing network is stable, which means that none of the queue lengths growarbitrarily large.

It is assumed that there are K stations in the network, numbered 1, 2, . . . , K. In addition,station 0 will represent the source of the external arrivals and station K + 1 will denotecustomers that leave the network. Furthermore σ

(k)j denotes the service time of the jth

customer served at station k, and ν(k)j denotes the station to which this customer is sent

after service is complete, i.e., the “next hop” in the network. We emphasize that in thismodel the service times and routing decisions are associated with the servers and not thecustomers, which is also true in the uncoordinated wireless network that we are studying.

Because of the stationarity and ergodicity assumptions, the long run averages of thesevariables are well defined. For the service times, we use the notation

1

n

n∑

j=1

σ(k)j →

1

µ(k)1 ≤ k ≤ K

2Recall that each node stores all packets that it receives, and therefore the physical buffer at each nodeonly increases in length. Because there is no feedback in the network layer protocol, nodes do not knowwhich of their transmissions were lost due to collisions. The queue length is thus a virtual quantity, of whichnodes do not have any knowledge. One consequence is that it is not possible to implement schemes wherethe transmission probability is modulated by queue length, such as [22, 23].

39

λ

Figure 4.10: A generalized Jackson network. The service times and routing decisions needonly be stationary and ergodic.

For the routing decisions, let Pi,j(n) =∑

ℓ≤n 1{ν

(i)ℓ

=j}where 0 ≤ i ≤ K and 1 ≤ j ≤ K + 1.

In words, Pi,j(n) counts how many of the first n customers served at station i are routed tostation j. Then, the averages of these variables converge to the following matrix

1

nPi,j(n) → pi,j 0 ≤ i ≤ K, 1 ≤ j ≤ K + 1

To find the stability conditions, we first must solve the following system of equations

πi = p0,i +K

∑

j=1

pj,iπj i = 1, . . . , K

The variable πi can be interpreted as the average number of times that a customer isprocessed by station i while it is in the network.

The stability of the network is defined via the time needed to empty all of the queuesin the network. The variable Z denotes the amount of time to empty the system after thelast external arrival, and A denotes the event that Z = ∞. The main result is stated interms of the probability of event A, as a function of the external arrival rate λ. This is mademathematically precise by considering a generalized Jackson network with only n externalarrivals. The time to empty this network after the last arrival is denoted by Z[−n,0], and Ais defined by the following limit

A ={

Z = limn→∞Z[−n,0] = ∞}

The stability conditions for the generalized Jackson network are given by the followingtheorem due to Lelarge [44], which is equivalent to earlier results due to Baccelli and Foss[45, 46]. We state the result without proof here.

Theorem 2. Letρ = λ max

1≤i≤K

πi

µ(i)

If ρ < 1 then P (A) = 0, while if ρ > 1 then P (A) = 1.

40

We note that this result gives a sharp threshold condition for stability. Depending onthe choice of the arrival rate λ, the network is either stable with probability one or unstablewith probability one.

Also, πi/µ(i) can be interpreted as the total service time of a customer at station i, i.e.,

the average number of times it visits the station multiplied by the average service time foreach visit. Thus, the theorem states that the arrival rate must be small enough that noneof the servers are overwhelmed. By maximizing over the servers, we are checking that eventhe worst station is not overwhelmed by arrival rate λ.

Stability Conditions of the Neighbor-Oblivious Wireless Network

To demonstrate the desired result, that the maximum throughput of the wireless networkis the min-cut of the equivalent wired network, we apply this theory of generalized Jacksonnetworks to the queuing network that we derived from the neighbor-oblivious wireless net-work.

We first describe the connection between stability of the queuing network and the rateof the wireless network. If the queueing network is stable, the queue length at each node isfinite and there are only a finite number of packets at all of the intermediate nodes at anygiven time. As time gets large, the rate of packets arriving at the destination must approachλ, the rate of packets injected into the network from the source. On the other hand, ifthe queueing network is unstable, one or more queue lengths becomes infinite and the rateof packets arriving at the destination is less than λ. Therefore, the maximum end-to-endthroughput is equal to the maximum rate λ for which the queueing network is stable.

Theorem 3: The generalized Jackson network derived from a rate allocation graph is stableif λ is less than the min-cut of the graph.

Proof of Theorem 3:The mean service rate of a node i, denoted by µ(i) in the above notation, is given by the

total capacity out of that node:

µ(i) =∑

e:out(i)

R∗i

Further, the transition probabilities pij are given by the fraction of outgoing rate fromnode i that is assigned to the link from i to j:

pij =Rij

∑

e:Out(i) Re

Thus, for any λ < λ∗, we can write the flow balance equation at node i as

∑

e:Out(i)

Re =∑

e:In(i)

Re

41

Defining Vi =∑

e:Out(i)

Re, we have

∑

e:In(i)

Re =∑

e=(j,i)

Re

Vj

Vj =∑

e=(j,i)

pjiVj = Vi

Thus, by setting Vi = λπi, we recover the same system of equations as in Theorem 2.Applying the theorem shows that

λπi

µ(i)=

Vi

µ(i)=

∑

e:Out(i) Re∑

e:Out(i) R∗e

= γ < 1

Therefore, Theorem 2 is satisfied and the network is stable. �

Combining Theorem 3 and Lemma 3 leads to the desired conclusion. Theorem 3 provesthat if we introduce a genie which drops packets so that the rate on the links is a valid flowwith value λ less than the min-cut of the equivalent wired network, then the destinationreceives a rate λ. Lemma 3 then shows that in the actual wireless network, the end-to-endthroughput must be at least as much as the throughput in this genie-degraded network.Therefore, any rate less than the min-cut of the equivalent wired network can be supported,even when the possibility of starvation is allowed. It is not necessary to assume that allnodes are in a saturated state.

42

Chapter 5

An Uncoordinated VANET

Communication Protocol

A completely uncoordinated communication scheme for a VANET, as opposed to thesynchronous scheme presented in Chapter 3 which required some coordination to spatiallypartition the vehicles, can be viewed as an instance of a neighbor-oblivious wireless network.We will refer to a network that uses the distributed MAC protocols outlined in Chapter 2 andrandom network coding, without any spatial synchronization, as an asynchronous scheme,in contrast to the synchronous scheme in Chapter 3.

We originally only required that the VANET protocol be distributed, while still allowingfor the possibility that vehicles could know, or learn, their neighbors. However, requiringthe VANET to operate in a neighbor-oblivious mode does not relax any of the originalconditions, and also prevents the use of feedback. This will simplify the implementation ofsuch a scheme in a practical deployment.

Theorem 3 in Chapter 4 shows that the capacity of the vehicular network is equal to themin-cut of the link capacity graph. The two main challenges when studying the VANETare (a) modeling the connectivity of the network, and (b) computing the min-cut of thelink capacity graph. Because the vehicles are no longer grouped into spatial blocks, a moreprecise model of the vehicle locations and connectivity is necessary.

The rest of this chapter is organized as follows. In Section 5.1 we present a model for theVANET topology. In Section 5.2 we calculate the end-to-end throughput of a dense vehicularnetwork under the slotted Aloha and unslotted Aloha MAC protocols, and discuss the op-timal centralized protocol. We show that the homogeneous connectivity of a dense networkenables data to be transmitted over large distances without a loss in performance. Finallyin Section 5.3 we give simulation results that compare the throughput of the asynchronousscheme under routing and network coding, and the throughput of the asynchronous schemeto the synchronous scheme proposed in Chapter 3

43

5.1 Modeling the Vehicular Network Connectivity

The transmission power used in vehicular ad hoc networks is typically quite large, whichin turn leads to a large transmission range. For example, transmitters operating in theDedicated Short Range Communication (DSRC) band can transmit up to a range of about300 meters [47]. Thus, the transmission range is significantly larger than the width of theroad, which is no more than about 50 meters for an 8 lane highway, and clearly much smallerfor a road with fewer lanes. Figure 5.1 shows the relative sizes of the transmission range androad width when the vehicles are making full use of the available transmit power.

R

Figure 5.1: The transmission range R of a vehicular network is typically much larger thanthe width of the road.

Because the transmission range is much larger than the width of the road, a reasonablemodel for the VANET connectivity is a one-dimensional line network, as shown in Figure5.2. Although the vehicles may be located in different lanes, we will make the approximationthat each vehicle can communicate with any other vehicles that are within a longitudinaldistance R from itself along the road1.

RR

Figure 5.2: The connectivity of the VANET will be modeled by a line network, where eachnode connects to all neighbors within a distance R of itself.

1When vehicles use a low transmit power, this approximation is no longer valid. In the case of extremelylow transmit power and a heavily congested road, where all lanes are at the maximum vehicle density, theconnectivity can be modeled as a two dimensional grid lattice. We leave the analysis of the throughput ofthe low power vehicular network to future work.

44

The number of neighbors of each node depends on the transmission range and the vehicledensity. An example where each node has 2 neighbors in both directions is shown in Figure5.3. A fixed number of neighbors for each node is an appropriate model for a heavilycongested highway, where the inter-vehicle spacing is constant. In this work, we focus onsuch a setting, and in particular quantify how the throughput scales as density increases. Fora road in the free-flow state, the distance between vehicles is not constant, e.g., the vehiclelocations may be modeled with a Poisson distribution. We will show in the following sectionthat the throughput is governed by the sparsest region of the network.

Figure 5.3: The connectivity of a VANET with evenly spaced nodes. Each vehicle has twoneighbors in both the forward and reverse directions.

5.2 Throughput of A Dense Vehicular Network

We begin by examining a VANET with constant vehicular spacing, such that there are Kedges emanating from each node, i.e., each vehicle can communicate with the next K vehiclesalong the road. In this section, we give exact expressions for the end-to-end throughputwith the slotted Aloha and unslotted Aloha MAC protocols, and an upper bound on thethroughput with a carrier sense MAC protocol.

We also make the additional assumption that the source node, which could be either avehicle or a fixed base station, has the ability to communicate with its neighbors withoutcontention. This could be implemented either by allowing local coordination at the sourcenode, or by allocating an auxiliary channel for the source to “seed” its own neighbors.Likewise, we also assume that the destination node can collect all of the packets received byits neighbors.

The motivation for these assumptions is to allow the unicast flow to take full advantage ofthe capacity provided by the multihop network. When the source and destination randomlyaccess the channel, similar to the other nodes, the min-cut of the link capacity graph willbe the cut that separates the source from the rest of the graph and the cut that separatesthe destination from the rest of the graph. Conceptually, the source cannot get informationinto the network fast enough to take advantage of the relays. By allowing the source anddestination to communicate without random access, the min-cut is not determined by thecuts in the boundary regions of the network. For simplicity, we will model the source anddestination as having infinite capacity links to their K neighbors2, as shown in Figure 5.4.

In this case, the minimum cut is given by any cut which does not pass through theseinfinite capacity links at the source and destination. Two such cuts are shown in Figure 5.5.

2In practice, these links will have finite capacity. However, since our objective is only to find the min-cutof the network, setting the capacity of links that do not pass through the minimum cut to be infinite willnot affect the result.

45

S D∞∞ ∞∞

Figure 5.4: A model of a vehicular network in which the source and destination can commu-nicate with their neighbors without randomly contending for the channel, as the relay nodesdo.

By the symmetry of the graph, all such cuts must have the same value. This proves ourfirst key result for the asynchronous transmission scheme: the end-to-end throughput of theVANET is independent of the number of relay nodes. Therefore, by using a random MACprotocol and network coding, data can be transmitted arbitrary distances along the road.Thus each vehicle can be viewed as a digital repeater

SCut 1 Cut 2

D∞∞ ∞∞

Figure 5.5: In a homogeneous vehicular network, all cuts that do not pass through the infinitecapacity edges near the source and destination will have the same (minimum) value.

We emphasize that this is not an intuitively obvious result. In the synchronous schemepresented in Chapter 3 the capacity is constant if the number of blocks grows subexponen-tially in the number of vehicles per block (the vehicle density), but decays to zero otherwise.In contrast to that result, the asynchronous scheme allows us to transmit data over arbitrarilylong distances while maintaining a constant throughput.

Our second result is to compute the numerical value of the capacity under the MACprotocols discussed in Section 2.2. We summarize the results in the limit of large density(large number of outgoing links K from every node) in Table 5.1, and discuss the computationin the following subsections.

46

MAC protocol CapacitySlotted Aloha 0.2248

Unslotted Aloha 0.1012TDMA 0.5

Table 5.1: The end-to-end throughput of the asynchronous transmission scheme, in the limitof high density, for three common MAC protocols.

5.2.1 Min-Cut of Vehicular Network Graph

We first note that under any MAC protocol, we must introduce virtual nodes to accountfor the broadcast nature of the wireless channel, as discussed in Section 4.3. Because themin-cut is finite, it must be true that none of the infinite capacity edges from the virtualnodes to the physical nodes pass through it3. Therefore, we must consider cuts of the formshown in Figure 5.6, where all of the virtual nodes that can be reached from nodes to theleft of the cut are on the right side of that cut.

Figure 5.6: The topology of the vehicular ad hoc network. The virtual nodes are representedby the dashed circles, and the infinite capacity edges from the virtual nodes to the physicalnodes by the bold lines. The minimum cut is a cut with no infinite capacity edges passingthrough it from the source side to the destination side.

5.2.2 Throughput under Slotted Aloha MAC Protocol

With slotted Aloha, the set of active links in any time slot is independent of all other timeslots. The Markov chain representing the network is then fully connected, with transitionprobabilities that depend only on the destination state. Thus, each link capacity can bedirectly computed by finding the probability that the link is active in a single time slot.

3Specifically, no infinite capacity edges pass through the min-cut from the source side of the graph to thedestination side.

47

We assume that each vehicle transmits with probability p in every time slot, independentof all other times and vehicles, where p is a parameter that will be optimized. We computethe sum of the values of all links crossing the cut shown in Figure 5.6. To simplify thecalculation, we will group together links by the “right most” node in the destination set. Asa concrete example, consider Figure 5.7 where each node can transmit to K = 2 neighbors.When node A transmits, the packet may be received by B only, C only, or both B and C.We simplify the computation of the cut by finding the sum of the link from A to C and thelink from A to the virtual node in one step. This sum is the probability that a transmissionfrom node A is received by node C, regardless of whether or not B receives it.

A B C

BC

Figure 5.7: To simplify computation of the min-cut of the vehicular network, the weight ofthe edge from A to C and the edge from A to the virtual node BC will be calculated in onestep.

After grouping the links in this manner, there are now two general classes of links crossingthe cut: groups that contain the “furthest neighbor” from the transmitter, and groups thatdo not reach the furthest neighbor. The total probability of each group in the first class isp(1−p)2K , which is the probability that the transmitting node chooses to access the channel(with probability p) and the furthest destination node and its remaining 2K − 1 neighborsare silent. There are K such links crossing the cut.

The probability of each group in the second class is p2(1− p)2K , because a second (inter-fering) node must also transmit to prevent the packet from being received by the furthestneighbor of the transmitter. The number of links in this class is given by

K−1∑

m=1

m =K(K − 1)

2

Therefore, the sum of the link capacities of all links crossing the cut is given by

Kp(1 − p)2K +K(K − 1)

2p2(1 − p)2K

≈ Kp(1 − p)2K +K2

2p2(1 − p)2K

48

By setting p = 12K

, the capacity in the limit of large K can be found using a standardapproximation as 1

21e+ 1

81e

= 58e

≈ 0.23. The capacity can be further increased by optimizingover p. However, numerical optimization shows that even for large K the improvement byusing the optimum p is relatively small, as shown in Figure 5.8.

0 5 10 15 20 25 30 35 40 45 500.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

Number of Neighbors (K)

Num

eric

al E

valu

atio

n of

Cut

Val

ue

Optimal pp = 1/(2K)

Figure 5.8: A numerical evaluation of the expression for the min-cut under slotted Aloha,for both the optimal transmission probability p and the nominal value p = 1

2K, reveals that

using the optimal value provides only a small improvement over the nominal value, even forlarge K.

To verify these results, we simulated the throughput in a network with the homoge-neous topology shown in Figure 5.4 using slotted Aloha and random network coding. First,we checked whether the end-to-end throughput does approach the predicted limit of 5

8eas

the number of neighbors of each node K increases. Figure 5.9 shows a simulation of thethroughput as a function of K. The total number of nodes between source and destinationwas varied, so that for all values of K, the expected number of hops was approximatelyfive4. Here, we simulated an idealized scenario, where network coding was done over the realfield. Thus, packets were only discarded by the receiver in the case when the transmitterwas starved of new information, and never because the transmitter had novel informationbut chose a bad set of mixing coefficients. Although in practical settings the coding must bedone over a finite field, by using a real field we were able to isolate the effect of starvationfrom the effect of random coding over a finite field. The plot shows that when K is on theorder of 20, the min-cut is already close to the limiting value of the high density regime.

4Observe that increasing K for fixed network size presents an unfair comparison, since each packet willtravel fewer hops between source and destination and thus the problem of starvation will be reduced.

49

Next, we simulated the dependence of the end-to-end throughput on the distance betweensource and destination. Figure 5.10 shows a plot of the simulated throughput as a function ofthe number of nodes between the source and destination, where the neighborhood size is fixedat K = 5 and the network coding is again done over the real field. This plot demonstratesthat the throughput is in fact nearly constant when the distance to the destination increases,as our analysis suggests.

2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

Sim

ulat

ed E

nd−

to−

End

Thr

ough

put

Number of Neighbors (K)

Figure 5.9: A simulation of the end-to-end throughput as a function of the neighborhoodsize K for the homogeneous topology shown in Fig 5.4. The network coding is done over thereal field.

5.2.3 Throughput under Unslotted Aloha MAC Protocol

We model the unslotted Aloha protocol with a system where each vehicle implementsslotted Aloha, but the slot boundaries are offset by a random phase. In this case, the sameanalysis as with the slotted Aloha MAC applies. However, when a transmitter sends a packet,the receiver and the interfering nodes must be silent for two consecutive time slots becauseof the offsets5. (See Figure 5.11 for a graphical example of the offset time slot boundaries.)Therefore, the capacity under unslotted Aloha is approximately equal to

Kp(1 − p)4K +K2

2p2(1 − p)4K

5We assume that whenever two transmissions collide for any portion of the packet duration, the entirepacket is lost.

50

10 15 20 25 30 35 40 45 50 55 600

0.05

0.1

0.15

0.2

0.25

Sim

ulat

ed E

nd−

to−

End

Thr

ough

put

Number of Nodes between Source and Destination

Figure 5.10: A simulation of the end-to-end throughput as a function of the number of nodesbetween the source and the destination for the homogeneous topology shown in Fig 5.4. Theneighborhood size is K = 5 and the coding is done over the real field.

By setting p = 14K

, the capacity can be analytically found to be equal to 932e

. As inthe case of slotted Aloha, numerical optimization shows that the capacity with the optimalprobability p is only slightly better.

5.2.4 Throughput under TDMA

While it is possible to write a general closed-form expression for the throughput of alinear network (a network where each node has K = 1 neighbors) using CSMA, evaluatingit is difficult. In addition, even finding such an expression for cases with K > 1 is complex.Thus, computing the end-to-end throughput of an uncoordinated VANET using CSMA isbeyond the scope of this dissertation.

However, in networks with no hidden nodes, it is known that a dual-channel CSMA sched-uler can achieve the same average throughput scaling law as the optimal TDMA scheduler[25]. Furthermore, a single-channel CSMA scheduler is known to achieve the same achiev-able rate region as the optimal TDMA scheduler, except for the boundary region [43, 22].For our purposes, this means that it is instructive to study the throughput of the VANETunder an optimal TDMA protocol, both because this provides an upper bound on whatan uncoordinated MAC can achieve and because CSMA can approach this bound in somesettings.

Consider the simple network where each node can reach K = 2 neighbors shown in Figure5.12. In order to optimally reuse the channel, in one time slot nodes 1, 6, 11, . . . will transmit

51

���������������

���������������

���������������

���������������

���������������

���������������

���������������

���������������

Figure 5.11: In unslotted Aloha, when a node transmits a packet (the shaded rectangle)neighboring nodes must be silent for two consecutive time slots (the crosshatched rectangles)to prevent a collision.

packets. In the second time slot, nodes 3, 8, 13, . . . will forward the packets that they receivedin the first slot. In each subsequent slot, packets are forwarded by the “furthest neighbor”that received the transmission in the previous time slot, with transmissions separated byjust enough distance to prevent any collisions. In a block of 2K +1 time slots, each node willreceive K novel packets (from nodes on its left), receive K redundant packets (from nodeson its right), and transmit once. Thus, for high densities the capacity of the optimal TDMAscheme approaches 1

2.

We note that with slotted Aloha, the throughput is about 45% of the throughput in afully coordinated network. Thus, the “cost of uncoordination” for slotted Aloha is a loss inperformance of about a factor of 2.

2 3 41 5 6 7

Figure 5.12: The optimal TDMA scheme is found by packing concurrent transmission astightly as possible without inducing collisions, and having each packet forwarded by thefurthest node that received it.

5.2.5 Throughput of Non-Homogeneous Networks

In practical settings, the vehicular network will not have a homogeneous connectivitystructure, even in relatively congested regimes. In lighter traffic conditions, the structurewill be even more variable. A first approximation to modeling such scenarios is to representthem by regions of constant density. Figure 5.13 shows an example with a more congestedregion on the left and a less congested region to the right.

Figure 5.13: To a first order, a non-homogeneous vehicular network can be approximated bya network with several different regions of constant connectivity.

52

In this case, the min-cut within each region is constant. The analysis in Section 5.2.2shows that the min-cut is an increasing function of K, the number of neighbors in a homo-geneous region. Thus, the smallest cut will be governed by the subsection of the VANETwith the least connectivity. (For systems with fixed transmit power, this will be the leastdense region, where the vehicles are spaced furthest apart.)

5.3 Simulation Results

We simulated the performance of the asynchronous scheme using the same parameters asthe synchronous simulations in Section 3.4. In these simulations, the asynchronous schemealways used the slotted Aloha MAC protocol. Our objective was to compare network cod-ing and routing in the asynchronous scheme, as well as to compare the asynchronous andsynchronous protocols. The main differences in the simulation parameters are the following:

• The nodes are not organized into spatial blocks, and are allowed to transmit in anytime slot.

• The time axis, although divided into slots as in the synchronous case, is not organizedinto frames.

• Each node chooses to transmit in every slot with probability p = 1/(L), where L is thenumber of neighbors within range of a node.

In the asynchronous scheme, there are no longer distinct waves of data flowing down thehighway. Thus, many of the practical considerations related to the length of packet headersand the field size, which are minimized in the synchronous case where nodes only combinepackets in the same wave, are much more severe. In particular, while the theoretical analysisassumes that vehicles can code over all source packets, this is not possible in even a moder-ately sized simulation, due to the fact that the packet headers would become unreasonablylong. There are thus nontrivial questions about when the infostation should transmit, andhow vehicles should separate packets that the infostation originally injected into the networkat different times. In these simulations, the source data is divided into finite length epochsinstead of coding over a single infinite block. In our simulations, we have assumed that theinfostation operates in exactly the same manner as in the synchronous case, transmittingfor N slots out of every 4N slots. The infostation protocol was kept unchanged in order tofacilitate a direct comparison between the synchronous and asynchronous schemes. However,the problem of identifying the optimal choice for the infostation transmission schedule in theasynchronous system is left for future work.

In addition, it was assumed that during each “duty cycle” the infostation would transmita new epoch of source data, independent of all previously transmitted epochs. When avehicle hears a packet from a new epoch, it immediately decodes the received packets fromthe last epoch and flushes them from its buffer. Although this policy of immediately flushing

53

the buffer, which was inspired by [27], is clearly far from optimal, it has the advantage ofbeing simple and robust6.

We first compared the performance of routing and network coding in the asynchronousprotocol. The first parameter we analyzed was the rank of the system as a function of thedistance from the infostation. Again, the rank is defined as the number of unique packetsper epoch for routing, and the number of independent packets per epoch for network coding.Figure 5.14 shows a plot of average rank versus the distance between the infostation and thedestination vehicle. The plot shows that in the asynchronous protocol the rank is constantas the distance increases when network coding is used, while with routing the rank decreasesrapidly to values close to 0.

In the case of network coding the asymptotic value of the curve is approximately equal toN/e independent packets per wave. Because the infostation initiates a new wave every 4Nslots in these simulations, the average throughput at the destination is thus about 1

4e. We

observe that this is less than 58e

, which is the throughput predicted by our analysis in Section5.2.2. However, the practical considerations discussed above clearly cause the performanceof the asynchronous scheme to degrade. In particular, the policy of immediately flushing thepackets in one epoch as soon as a vehicle receives a single packet in the next epoch causesthe throughput to decrease. The design of network protocols which can be implemented inpractice and bridge this gap is an important open question.

As in the case of the synchronous protocol, we again measured the total number ofreceived packets, regardless of whether they were independent. Figure 5.15 confirms thatthere is no difference in the number of raw received packets with the two strategies.

Finally, we also empirically studied the delay introduced by the asynchronous scheme.Figure 5.16 shows the time when the infostation begins transmitting the first packet of eachepoch, as well as the time when the destination node receives the last packet of that epoch.The results are shown only for network coding. In the asynchronous scheme, the latencybefore the first wave is completed is very low, about 0.7 s.

We also compared the performance of the synchronous and asynchronous schemes. Fig-ure 5.17 compares the number of independent packets received at the destination vehicle.In this figure the results obtained in the case of synchronous strategies without overhearingwere not considered, since it was empirically shown in Section 3.4 that incorporating over-hearing provides better performance without any additional cost. The plot shows that thesynchronous scheme can sustain higher rank than the asynchronous scheme, with an increaseon the order of 50%. While the asynchronous network coding scheme does not provide thesame throughput as the synchronous network coding scheme, we still would like to highlightthe fact that the asynchronous scheme can sustain a constant throughput over relativelylarge distances, without the synchronous block structure. Similar considerations hold inthe case of routing, however neither the synchronous nor asynchronous schemes can sustainan acceptable throughput over long distances. The fact that the synchronous scheme out-performs the asynchronous scheme is to be expected, since the spatial partitioning imposesadditional structure on the problem. Because additional resources must be invested to set

6It is not possible for vehicles to combine packets across source epochs, since the overhead required totransmit the coefficients would eventually consume the entire packet

54

0 1000 2000 3000 4000 5000 6000 7000 80000

20

40

60

80

100

120

Distance (m)

Ran

k pe

r w

ave

Asynchronous RoutingAsynchronous Network Coding

Figure 5.14: The number of independent packets received per epoch as a function of thelocation of the destination vehicle in the asynchronous transmission scheme. Results areshown for two strategies: routing with the asynchronous scheme and network coding withthe asynchronous scheme.

up the spatial blocks, it is unsurprising that better performance is obtained.On the other hand, the results for the delay, presented in Figure 5.18, indicate that the

asynchronous strategy can guarantee a lower initial latency, due to the pipeline effect thatis artificially introduced in the synchronous scheme, which forces nodes to wait for a longertime before the first wave is received. The figure shows indeed that there is a trade-offbetween throughput and latency in the selection of a synchronous or asynchronous method.If the main goal is to guarantee a high throughput and the system has the resources set upthe spatial block structure, then the synchronous solution should be used, at the price of ahigher complexity and a larger initial latency. On the other hand, if the throughput providedby asynchronous network coding is sufficient, then this strategy is preferable because of thelower complexity and the lower latency.

55

0 1000 2000 3000 4000 5000 6000 7000 8000120

140

160

180

200

220

240

Distance (m)

Rec

eive

d pa

cket

s pe

r w

ave

Asynchronous RoutingAsynchronous Network Coding

Figure 5.15: The total number of received packets per epoch as a function of the locationof the destination vehicle in the asynchronous transmission scheme. This plot counts allpackets received without collision, regardless of whether they are independent. Results areshown for two strategies: routing with the asynchronous scheme and network coding withthe asynchronous scheme.

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1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Wave

Tim

e (s

)

Start of TransmissionCompletion Time for Asynchronous Network Coding

Figure 5.16: The time when the infostation begins transmitting each epoch, and the timewhen the destination vehicle receives the final packet of the epoch, for the asynchronousscheme. The vertical distance between the two curves is the time delay of the asynchronousscheme with network coding.

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0 1000 2000 3000 4000 5000 6000 7000 80000

20

40

60

80

100

120

Distance (m)

Ran

k pe

r w

ave

Synchronous Routing + OverhearingSynchronous Network Coding + OverhearingAsynchronous RoutingAsynchronous Network Coding

Figure 5.17: The number of independent packets received per wave as a function of thelocation of the destination vehicle. Results are shown for four strategies: routing with thesynchronous overhearing scheme, network coding with the synchronous overhearing scheme,routing with the asynchronous scheme, and network coding with the asynchronous scheme.

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1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

Wave

Tim

e (s

)

Start of TransmissionCompletion Time for Synchronous Network Coding + OverhearingCompletion Time for Asynchronous Network Coding

Figure 5.18: The time when the infostation begins transmitting each wave, and the timewhen the destination vehicle receives the final packet of the wave for network coding withthe synchronous overhearing scheme and network coding with the asynchronous scheme. Thevertical distance between the curves is the time delay.

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Chapter 6

Conclusions and Future Directions

In this dissertation, we have considered the problem of multihop communication in ve-hicular ad hoc networks. Because vehicles frequently enter and exit roads, and differentvehicles may be traveling at very different speeds, the topology of VANETs changes rapidly.Therefore, communication protocols that rely on coordinated scheduling of transmissionsto avoid collisions and global routing tables to move data from source to destination areeither technically infeasible, or require that a very large portion of system resources by de-voted to tracking and disseminating the network topology. We have proposed to instead userandomized, uncoordinated protocols in such networks.

We have proposed to combine randomized MAC protocols, such as Aloha and a simplecarrier sensing protocol, with random linear network coding for use in VANETs. Networkcoding, which is based on the mixing of packets at intermediate nodes, makes the systemrobust to packet losses without requiring any link level feedback. Our work focuses on analyz-ing the throughput that is provided by vehicular networks that implement this combinationof MAC and network layer protocols.

In particular, we have focused on the case of a single multihop unicast flow, where the ve-hicles on the road are serving as relays to forward packets from the source to the destination.We began by considering a spatially synchronized protocol. This scheme required a minordegree of coordination, in that vehicles are grouped into spatial blocks and transmissionsare scheduled at the block level to move waves of data down the road, although vehicleswithin each block randomly contend with each other when transmitting. We proved thatthe throughput with network coding in this synchronous scheme is constant, as long as thenumber of blocks grows slower than exponentially in the vehicle density.

We then turned to a completely uncoordinated protocol. We showed that when vehiclescombine random channel access with network coding, the throughput is given by the min-cut of the link capacity graph between source and destination. The implication is that on acongested highway, where the connectivity is homogeneous and every vehicle has the samenumber of neighbors, the throughput does not degrade with distance. In contrast to thespatially synchronized protocol, a constant throughput can be supported over an arbitrarilylong distance.

The key challenge in demonstrating this result is the problem of starvation. Due to therandom ordering of transmissions induced by the randomized MAC protocol, the intermedi-

60

ate nodes may temporarily not have any new information to forward to their neighbors. Weused tools from queueing theory to show that starvation does not decrease the throughputbelow the min-cut of the network. This result has many applications beyond the vehicularnetworks considered here. It closes an open question in the analysis of CSMA networks,where it is often assumed that all nodes are in a saturated state, meaning that they alwayshave new information to transmit when they access the channel.

There are numerous open questions and directions for future research, both theoreticaland applied, that must be addressed in order to make VANETs a practical reality in thenear future:

• In practice, in order for the destination to be able to decode its received packets, eachpacket must contain a header that specifies the linear combination contained in itsbody. Therefore, vehicles cannot code over an infinite number of source packets. Thisis especially true in wireless networks, where the packet length must be smaller thanthe coherence time of the channel, and thus the header must be kept to a reasonablysmall length. Due to these practical issues, the source data is partitioned into epochs,and intermediate nodes only combine packets in the same epoch. Before deployinga system that incorporates network coding, the performance loss that results fromdividing the source data into epochs must be analyzed. In particular, the effect ofpartitioning the source on the result that a constant throughput can be sent over anarbitrary number of hops must be carefully studied.

• While the problem of sending a single unicast flow through a vehicular network is animportant starting point, a practical network must be able to support a variety of othermodes of communication, such as multicast. One of the most interesting problems iswhen there are multiple overlapping streams. Future work must address how receiversthat are interested in different flows should share the channel and cooperate with eachother in relaying packets.

• In this work, we have focused on data streams which are short enough that the networkcan be considered effectively static during the duration of transmission. For very longdata streams, such as movies, the vehicles will move relative to each other while thedata is being relayed, which can improve performance by physically carrying datathrough regions of low connectivity but presents additional challenges to quantifyingnetwork performance.

• We have up to this point considered a simplified physical layer model, where receiverswithin range of a transmitter can perfectly demodulate a packet, and receivers out ofrange experience no interference. A more accurate analysis of the system performanceshould incorporate a more realistic physical layer model based on the SINR at thereceiver.

• A more accurate approximation of the individual link capacities under the CSMAprotocol, as a function of the number of neighbors of each vehicle, is worth exploring.

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We hope that the results contained in this dissertation will prompt further research in thefield of protocol design and analysis for vehicular ad hoc networks, and eventually lead to thedeployment of practical networks which can serve to enable intelligent transportation systemsthat improve highway safety and reduce the expenses associated with traffic congestion.

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